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General complex Lp projection bodies and complex Lp mixed projection bodies

General complex Lp projection bodies and complex Lp mixed projection bodies 1IntroductionThe classical Brunn-Minkowski theory appeared at the turn of the nineteenth into the twentieth century. One of the core concepts that Minkowski introduced within the Brunn-Minkowski theory is the projection body. There are important inequalities involving the volume of the projection body and its polar, such as Petty projection inequality [1] and Zhang projection inequality [2].In the early 1960s, Firey [3] introduced the Firey-Minkowski Lp{L}_{p}-addition of a convex body. In the mid-1990s, it was shown in [4] and [5] that a study of the volume of the Firey-Minkowski Lp{L}_{p}combinations leads to the Lp{L}_{p}Brunn-Minkowski theory. This theory has expanded rapidly. An early achievement of the Lp{L}_{p}Brunn-Minkowski theory was the discovery of Lp{L}_{p}projection body, introduced by Lutwak et al. [6]. Since then, Ludwig [7,8] extended the projection body to an entire class that can be called the general Lp{L}_{p}projection body.A mixed projection body was introduced in the classic volume of Bonnesen-Fenchel [9]. In [10] and [11], Lutwak considered the volume of the mixed projection body and established the classical mixed volume inequalities, such as Aleksandrov-Fenchel inequalities and Brunn-Minkowski inequalities.Let us mention that the projection bodies described above are all real. The theory of the real projection body continues to be a very active field now. For additional information and some results on real projection body see, e.g., [8,12, 13,14,15, 16,17]. However, some classical concepts of convex geometry in real vector space were extended to complex cases, such as complex difference body [18], complex intersection body [19], complex centroid body [20,21], and complex projection body [22,23, 24,25].In this paper, we mainly study the projection body in complex vector space. Let K(Cn){\mathcal{K}}\left({{\mathbb{C}}}^{n})be the set of convex body (nonempty compact convex subsets) in complex vector space Cn{{\mathbb{C}}}^{n}. For the set of the convex body containing the origin in their interiors and the set of an origin-symmetric convex body in Cn{{\mathbb{C}}}^{n}, we write Ko(Cn){{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n})and Kos(Cn){{\mathcal{K}}}_{os}\left({{\mathbb{C}}}^{n}), respectively. Let V(K)V\left(K)denote the complex volume of KK, BBthe complex unit ball, and S2n−1{S}^{2n-1}the complex unit sphere.In 2011, the complex Lp{L}_{p}projection body ΠCK{\Pi }_{C}Kwas defined by Abardia and Bernig [22]: For K∈K(Cn)K\in {\mathcal{K}}\left({{\mathbb{C}}}^{n})and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}}), (1.1)h(ΠCK,u)=nV1(K,C⋅u)=12∫S2n−1h(C⋅u,v)dSK(v)h\left({\Pi }_{C}K,u)=n{V}_{1}\left(K,C\cdot u)=\frac{1}{2}\mathop{\int }\limits_{{S}^{2n-1}}h\left(C\cdot u,v){\rm{d}}{S}_{K}\left(v)\hspace{1.6em}for every u∈S2n−1u\in {S}^{2n-1}, where C⋅u≔{cu:c∈C}C\cdot u:= \{cu:c\in C\}. They also defined the mixed complex projection body as (1.2)h(ΠC(K1,K2,…,K2n−1),u)=nV(K1,K2,…,K2n−1,C⋅u).h\left({\Pi }_{C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),u)=nV\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1},C\cdot u).Very recently, the concept of asymmetric complex Lp{L}_{p}projection body Πp,C+K{\Pi }_{p,C}^{+}Kwas introduced in [24]. First, a convex body C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})is called an asymmetric Lp{L}_{p}zonoid if there exists a finite even Borel measure μp,C(v){\mu }_{p,C}\left(v)on Sn−1{S}^{n-1}such that hC(u)p=∫Sn−1(ℛ[cu⋅v])+pdμp,C(c).{h}_{C}{\left(u)}^{p}=\mathop{\int }\limits_{{S}^{n-1}}{\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{+}^{p}{\rm{d}}{\mu }_{p,C}\left(c).Based on the fact that hC⋅u(v)=hC(u⋅v){h}_{C\cdot u}\left(v)={h}_{C}\left(u\cdot v)and the sesquilinearity of the Hermitian inner product in Cn{{\mathbb{C}}}^{n}, we obtain hC⋅u(v)p=∫Sn−1(ℛ[c⋅(u⋅v)])+pdμp,C(c)=∫Sn−1(ℛ[cu⋅v])+pdμp,C(c){h}_{C\cdot u}{\left(v)}^{p}=\mathop{\int }\limits_{{S}^{n-1}}{\left({\mathcal{ {\mathcal R} }}\left[c\cdot \left(u\cdot v)])}_{+}^{p}{\rm{d}}{\mu }_{p,C}\left(c)=\mathop{\int }\limits_{{S}^{n-1}}{\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{+}^{p}{\rm{d}}{\mu }_{p,C}\left(c)\hspace{1.75em}for all u,v∈Sn−1u,v\in {S}^{n-1}. Then, if p≥1,K∈Ko(Cn)p\ge 1,K\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})is an asymmetric Lp{L}_{p}zonoid, the asymmetric complex Lp{L}_{p}projection body Πp,C+K{\Pi }_{p,C}^{+}Kis (1.3)h(Πp,C+K,u)p=2nVp(K,C⋅u)=∫S2n−1∫Sn−1(ℛ[cu⋅v])+pdμp,C(c)dSp,K(v)h{\left({\Pi }_{p,C}^{+}K,u)}^{p}=2n{V}_{p}\left(K,C\cdot u)=\mathop{\int }\limits_{{S}^{2n-1}}\mathop{\int }\limits_{{S}^{n-1}}{\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{+}^{p}{\rm{d}}{\mu }_{p,C}\left(c){\rm{d}}{S}_{p,K}\left(v)for all u,v∈S2n−1u,v\in {S}^{2n-1}, where Sp,K(v){S}_{p,K}\left(v)is the Lp{L}_{p}surface area measure of KKon S2n−1{S}^{2n-1}.Motivated by the works of Abardia and Bernig [22], Haberl [20], and Liu and Wang [24], we introduce more general definitions of complex Lp{L}_{p}projection body and complex Lp{L}_{p}mixed projection body.Definition 1.1If p≥1p\ge 1, K∈Ko(Cn)K\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})is an asymmetric Lp{L}_{p}zonoid, the general complex Lp{L}_{p}projection body Πp,CλK{\Pi }_{p,C}^{\lambda }Kis defined by (1.4)h(Πp,CλK,u)p=f1(λ)h(Πp,C+K,u)p+f2(λ)h(Πp,C−K,u)ph{\left({\Pi }_{p,C}^{\lambda }K,u)}^{p}={f}_{1}\left(\lambda )h{\left({\Pi }_{p,C}^{+}K,u)}^{p}+{f}_{2}\left(\lambda )h{\left({\Pi }_{p,C}^{-}K,u)}^{p}for every λ∈[−1,1]\lambda \in \left[-1,1], where f1(λ)=(1+λ)p(1+λ)p+(1−λ)p,f2(λ)=(1−λ)p(1+λ)p+(1−λ)p,{f}_{1}\left(\lambda )=\frac{{\left(1+\lambda )}^{p}}{{\left(1+\lambda )}^{p}+{\left(1-\lambda )}^{p}},\hspace{1em}{f}_{2}\left(\lambda )=\frac{{\left(1-\lambda )}^{p}}{{\left(1+\lambda )}^{p}+{\left(1-\lambda )}^{p}},and f1(λ)+f2(λ)=1{f}_{1}\left(\lambda )+{f}_{2}\left(\lambda )=1.We use Πp,Cλ,∗K{\Pi }_{p,C}^{\lambda ,\ast }Kto denote the polar body Πp,CλK{\Pi }_{p,C}^{\lambda }K. The normalization is chosen such that Πp,CλB=B{\Pi }_{p,C}^{\lambda }B=Band Πp,C0K=Πp,CK{\Pi }_{p,C}^{0}K={\Pi }_{p,C}K. If λ=1\lambda =1in (1.4), then Πp,C1K=Πp,C+K{\Pi }_{p,C}^{1}K={\Pi }_{p,C}^{+}K. In addition, set Πp,C−K=Πp,C+(−K){\Pi }_{p,C}^{-}K={\Pi }_{p,C}^{+}\left(-K). By the definitions of Πp,C±K{\Pi }_{p,C}^{\pm }Kand Πp,CλK{\Pi }_{p,C}^{\lambda }K, we obtain (1.5)Πp,CλK=(1+λ)p(1+λ)p+(1−λ)p⋅Πp,C+K+p(1−λ)p(1+λ)p+(1−λ)p⋅Πp,C−K,{\Pi }_{p,C}^{\lambda }K=\frac{{\left(1+\lambda )}^{p}}{{\left(1+\lambda )}^{p}+{\left(1-\lambda )}^{p}}\cdot {\Pi }_{p,C}^{+}K{+}_{p}\frac{{\left(1-\lambda )}^{p}}{{\left(1+\lambda )}^{p}+{\left(1-\lambda )}^{p}}\cdot {\Pi }_{p,C}^{-}K,and the complex Lp{L}_{p}projection body is defined as (1.6)Πp,CK=12⋅Πp,C+K+p12⋅Πp,C−K.{\Pi }_{p,C}K=\frac{1}{2}\cdot {\Pi }_{p,C}^{+}K{+}_{p}\frac{1}{2}\cdot {\Pi }_{p,C}^{-}K.It is clear that if p=1p=1in (1.6), Πp,CK{\Pi }_{p,C}Kis ΠCK{\Pi }_{C}Kdefined in (1.1).Definition 1.2If p≥1p\ge 1, K1,K2,…,K2n−1∈Ko(Cn){K}_{1},{K}_{2},\ldots ,{K}_{2n-1}\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})is an asymmetric Lp{L}_{p}zonoid, the general complex Lp{L}_{p}mixed projection body Πp,Cλ(K1,K2,…,K2n−1){\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})is defined by (1.7)h(Πp,Cλ(K1,K2,…,K2n−1),u)p=f1(λ)h(Πp,C+(K1,K2,…,K2n−1),u)p+f2(λ)h(Πp,C−(K1,K2,…,K2n−1),u)p\hspace{1.8em}h{\left({\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),u)}^{p}={f}_{1}\left(\lambda )h{\left({\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),u)}^{p}+{f}_{2}\left(\lambda )h{\left({\Pi }_{p,C}^{-}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),u)}^{p}for every λ∈[−1,1]\lambda \in \left[-1,1].Moreover, if K2n−i=⋯=K2n−1=B{K}_{2n-i}=\cdots ={K}_{2n-1}=Band M≔(K1,K2,…,K2n−1−i){\bf{M}}:= \left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1-i}), then for 0≤i≤2n−20\le i\le 2n\hspace{-0.02em}-\hspace{-0.02em}2, Πp,Cλ(K1,K2,…,K2n−1){\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})is written as Πp,i,Cλ(M){\Pi }_{p,i,C}^{\lambda }\left({\bf{M}}). If K1=⋯=K2n−1−i=K{K}_{1}=\cdots ={K}_{2n-1-i}=K, we simply write Πp,i,Cλ(M){\Pi }_{p,i,C}^{\lambda }\left({\bf{M}})as Πp,i,Cλ(K){\Pi }_{p,i,C}^{\lambda }\left(K)that is called the iith Lp{L}_{p}mixed complex projection body of KK. If i=0i=0, we write Πp,0,CλK{\Pi }_{p,0,C}^{\lambda }Kas Πp,CλK{\Pi }_{p,C}^{\lambda }K.Before stating our main results, let us introduce the Lp{L}_{p}Blaschke combination. For 2n≠p≥12n\ne p\ge 1and K,L∈Kos(Cn)K,L\in {{\mathcal{K}}}_{os}\left({{\mathbb{C}}}^{n}), the Lp{L}_{p}Blaschke combination K#pL∈Kos(Cn)K\hspace{-0.3em}{\text{\#}}_{p}L\in {{\mathcal{K}}}_{os}\left({{\mathbb{C}}}^{n})is defined by Lutwak [4] as (1.8)dSp(K#pL,⋅)≔dSp(K,⋅)+dSp(L,⋅),{\rm{d}}{S}_{p}\left(K\hspace{-0.3em}{\text{\#}}_{p}L,\cdot ):= {\rm{d}}{S}_{p}\left(K,\cdot )+{\rm{d}}{S}_{p}\left(L,\cdot ),where Sp(K,⋅){S}_{p}\left(K,\cdot )denotes the Lp{L}_{p}surface area measure of KKon S2n−1{S}^{2n-1}. If p=1p=1and K,L∈K(Cn)K,L\in {\mathcal{K}}\left({{\mathbb{C}}}^{n}), it is a classical Blaschke combination.Our main results can be stated as the following Theorems 1.1–1.4 and among them, Theorems 1.1–1.2 are the Brunn-Minkowski-type inequalities for the general complex Lp{L}_{p}projection bodies. Theorems 1.3–1.4 are the Aleksandrov-Fenchel-type inequalities for the general complex Lp{L}_{p}mixed projection bodies.Theorem 1.1If p≥1p\ge 1, K,L∈Kos(Cn)K,L\in {{\mathcal{K}}}_{os}\left({{\mathbb{C}}}^{n}), and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})is an asymmetric Lp{L}_{p}zonoid, then(1.9)V(Πp,Cλ(K#pL))p2n≥V(Πp,CλK)p2n+V(Πp,CλL)p2nV{\left({\Pi }_{p,C}^{\lambda }\left(K{\text{\#}}_{p}L))}^{\tfrac{p}{2n}}\ge V{\left({\Pi }_{p,C}^{\lambda }K)}^{\tfrac{p}{2n}}+V{\left({\Pi }_{p,C}^{\lambda }L)}^{\tfrac{p}{2n}}with equality if and only if KKand LLare real dilates.Theorem 1.2If p≥1p\ge 1, K,L∈Kos(Cn)K,L\in {{\mathcal{K}}}_{os}\left({{\mathbb{C}}}^{n}), and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})is an asymmetric Lp{L}_{p}zonoid, then(1.10)V(Πp,Cλ,∗(K#pL))−p2n≥V(Πp,Cλ,∗K)−p2n+V(Πp,Cλ,∗L)−p2nV{\left({\Pi }_{p,C}^{\lambda ,\ast }\left(K{\text{\#}}_{p}L))}^{-\tfrac{p}{2n}}\ge V{\left({\Pi }_{p,C}^{\lambda ,\ast }K)}^{-\tfrac{p}{2n}}+V{\left({\Pi }_{p,C}^{\lambda ,\ast }L)}^{-\tfrac{p}{2n}}with equality if and only if KKand LLare real dilates.Theorem 1.3If p≥1p\ge 1, K1,K2,…,K2n−1∈Ko(Cn){K}_{1},{K}_{2},\ldots ,{K}_{2n-1}\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})is an asymmetric Lp{L}_{p}zonoid, then(1.11)V(Πp,Cλ(K1,K2,…,K2n−1))pr2n≥(2n)1−p∏j=12n−1V(K1,…,K2n−1,Kj)r(1−p)2n−1∏j=1rV(ΠC(Kj[r],Kr+1,…,K2n−1))p2n.V{\left({\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{pr}{2n}}\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{r\left(1-p)}{2n-1}}\mathop{\prod }\limits_{j=1}^{r}V{\left({\Pi }_{C}\left({K}_{j}\left[r],{K}_{r+1},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n}}.If Kj{K}_{j}is an ellipsoid centered at the origin or an Hermitian ellipsoid, the equality holds.Theorem 1.4If K1,K2,…,K2n−1−i∈Ko(Cn){K}_{1},{K}_{2},\ldots ,{K}_{2n-1-i}\hspace{-0.03em}\in \hspace{-0.03em}{{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n})and C∈K(C)C\hspace{-0.03em}\in \hspace{-0.03em}{\mathcal{K}}\left({\mathbb{C}})is an asymmetric Lp{L}_{p}zonoid. Let M≔(K1,K2,…,K2n−1−i){\bf{M}}\hspace{-0.03em}:= \hspace{-0.03em}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1-i}). If p≥1p\ge 1, i=0,1,…,2n−2i=0,1,\ldots ,2n-2, then(1.12)V(Πp,i,CλM)p2n≥(2n)1−p∏j=12n−1−iWi(M,Kj)1−p2n−1−i∏j=1rV(Πi,C(Kj[r],Kr+1,…,K2n−1−i))p2nr.V{\left({\Pi }_{p,i,C}^{\lambda }{\bf{M}})}^{\tfrac{p}{2n}}\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1-i}{W}_{i}{\left({\bf{M}},{K}_{j})}^{\tfrac{1-p}{2n-1-i}}\mathop{\prod }\limits_{j=1}^{r}V{\left({\Pi }_{i,C}\left({K}_{j}\left[r],{K}_{r+1},\ldots ,{K}_{2n-1-i}))}^{\tfrac{p}{2nr}}.If Kj{K}_{j}is an ellipsoid centered at the origin or an Hermitian ellipsoid, the equality holds.Remark 1.1Note that the cases of C=[0,1]C=\left[0,1]of Theorems 1.1–1.4 are the Brunn-Minkowski-type inequalities [26] and the Aleksandrov-Fenchel-type inequalities for the general Lp{L}_{p}projection bodies [27].If CCis just a point, then Πp,CλK={0}{\Pi }_{p,C}^{\lambda }K=\{0\}for every K∈Ko(Cn)K\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n})and every λ∈[−1,1]\lambda \in \left[-1,1]. We assume that dimC>0\hspace{0.1em}\text{dim}\hspace{0.1em}\hspace{0.33em}C\gt 0throughout this paper.2Preliminaries2.1Support function, radial function, and polar of convex bodyFor a complex number c∈Cc\in {\mathbb{C}}, we write c¯\bar{c}for its complex conjugate and ∣c∣| c| for its norm. If ϕ∈Cm×n\phi \in {{\mathbb{C}}}^{m\times n}, then ϕ∗{\phi }^{\ast }denotes the conjugate transpose of ϕ\phi . We denote by “⋅\cdot ” the standard Hermitian inner product on Cn{{\mathbb{C}}}^{n}which is conjugate linear in first argument. Koldobsky et al. [19] identified Cn{{\mathbb{C}}}^{n}with R2n{{\mathbb{R}}}^{2n}using the standard mapping: ι(c)=(ℛ[c1],…,ℛ[cn],ζ[c1],…,ζ[cn]),c∈Cn,\iota \left(c)=\left({\mathcal{ {\mathcal R} }}\left[{c}_{1}],\ldots ,{\mathcal{ {\mathcal R} }}\left[{c}_{n}],\zeta \left[{c}_{1}],\ldots ,\zeta \left[{c}_{n}]),\hspace{1em}c\in {{\mathbb{C}}}^{n},\hspace{1.35em}where ℛ{\mathcal{ {\mathcal R} }}and ζ\zeta are the real and imaginary parts, respectively. Note that ℛ[x⋅y]=ιx⋅ιy{\mathcal{ {\mathcal R} }}\left[x\cdot y]=\iota x\cdot \iota yfor all x,y∈Cnx,y\in {{\mathbb{C}}}^{n}, where the inner product on the right hand is the standard Euclidean inner product on R2n{{\mathbb{R}}}^{2n}.We collect complex reformulations of well-known results from convex geometry. These complex version can be directly deduced from their real counterparts by an appropriate application of ι\iota . For more details refer to the books in [28,29].A convex body K∈K(Cn)K\in {\mathcal{K}}\left({{\mathbb{C}}}^{n})is determined by its support function hK:Cn→R{h}_{K}:{{\mathbb{C}}}^{n}\to {\mathbb{R}}, where hK(x)=max{ℛ[x⋅y]:y∈K}.{h}_{K}\left(x)=\hspace{0.1em}\text{max}\hspace{0.1em}\{{\mathcal{ {\mathcal R} }}\left[x\cdot y]:y\in K\}.For every Borel set ω⊂S2n−1\omega \subset {S}^{2n-1}, the complex surface area measure SK{S}_{K}of K∈K(Cn)K\in {\mathcal{K}}\left({{\mathbb{C}}}^{n})is defined by SK(ω)=ℋ2n−1(ι{x∈K:∃u∈ω,ℛ[x⋅u]=hK(u)}),{S}_{K}\left(\omega )={{\mathcal{ {\mathcal H} }}}^{2n-1}\left(\iota \{x\in K:\exists u\in \omega ,{\mathcal{ {\mathcal R} }}\left[x\cdot u]={h}_{K}\left(u)\}),where ℋ2n−1{{\mathcal{ {\mathcal H} }}}^{2n-1}stands for (2n−12n-1)-dimensional Hausdorff measure on R2n{{\mathbb{R}}}^{2n}. In addition, the complex surface area measures are translation invariant and ScK(ω)=SK(c¯ω){S}_{cK}\left(\omega )={S}_{K}\left(\bar{c}\omega )for all c∈Sn−1c\in {S}^{n-1}and each Borel set ω∈S2n−1\omega \in {S}^{2n-1}.KKis an Hermitian ellipsoid if K={x∈Cn:x⋅ϕx≤1}+tK=\left\{x\in {{\mathbb{C}}}^{n}:x\cdot \phi x\le 1\right\}+tfor a positive definite Hermitian matrix ϕ∈GL(n,C)\phi \in GL\left(n,{\mathbb{C}})and a t∈Cnt\in {{\mathbb{C}}}^{n}. Note that if KKis an Hermitian ellipsoid if and only if K=ψB+tK=\psi B+tfor some ψ∈GL(n,C)\psi \in GL\left(n,{\mathbb{C}})and a t∈Cnt\in {{\mathbb{C}}}^{n}(see [20]).If KKis a compact star-shaped (about the origin) in Cn{{\mathbb{C}}}^{n}, its radial function, ρK(x)=ρ(K,x):Cn⧹{0}→[0,∞){\rho }_{K}\left(x)=\rho \left(K,x):{{\mathbb{C}}}^{n}\setminus \{0\}\to {[}0,\infty )is given by ρK(x)=max{λ≥0:λx∈K}.{\rho }_{K}\left(x)=\hspace{0.1em}\text{max}\hspace{0.1em}\{\lambda \ge 0:\lambda x\in K\}.If ρK(x){\rho }_{K}\left(x)is positive and continuous, KKwill be called a star body. For the set of star body containing the origin in their interiors, we write So(Cn){{\mathcal{S}}}_{o}\left({{\mathbb{C}}}^{n}). Moreover, if K∈Ko(Cn)K\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), then K∗∈Ko(Cn){K}^{\ast }\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}). On Cn⧹{0}{{\mathbb{C}}}^{n}\setminus \{0\}, the support function and radial function of the polar body K∗{K}^{\ast }can be given, respectively, by hK∗=1ρK,ρK∗=1hK.{h}_{{K}^{\ast }}=\frac{1}{{\rho }_{K}},\hspace{1em}{\rho }_{{K}^{\ast }}=\frac{1}{{h}_{K}}.\hspace{3.2em}2.2The Lp{L}_{p}mixed quermassintegralsFor K∈Ko(Cn)K\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), i=1,2,…,2n−1i=1,2,\ldots ,2n-1, the quermassintegrals Wi(K){W}_{i}\left(K)of KKare defined by (see [29]) (2.1)Wi(K)=12n∫S2n−1h(K,u)dSi(K,u),{W}_{i}\left(K)=\frac{1}{2n}\mathop{\int }\limits_{{S}^{2n-1}}h\left(K,u){\rm{d}}{S}_{i}\left(K,u),where Si(K,u){S}_{i}\left(K,u)is the mixed surface area of KK.If p≥1,K,L∈Ko(Cn)p\ge 1,K,L\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), and α,β≥0\alpha ,\beta \ge 0that are not both zero, the Lp{L}_{p}Minkowski combination α⋅K+pβ⋅L\alpha \cdot K{+}_{p}\beta \cdot Lis defined by hα⋅K+pβ⋅Lp=αhKp+βhLp.{h}_{\alpha \cdot K{+}_{p}\beta \cdot L}^{p}=\alpha {h}_{K}^{p}+\beta {h}_{L}^{p}.The extension of the Brunn-Minkowski inequality (see [4]) is as follows: If K,L∈Ko(Cn),p>1K,L\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}),p\gt 1, then (2.2)Wi(K+pL)p2n−i≥Wi(K)p2n−i+Wi(L)p2n−i{W}_{i}{\left(K{+}_{p}L)}^{\tfrac{p}{2n-i}}\ge {W}_{i}{\left(K)}^{\tfrac{p}{2n-i}}+{W}_{i}{\left(L)}^{\tfrac{p}{2n-i}}with equality if and only if KKand LLare dilates. In particular, if p=1p=1, i=0i=0, the inequality is the Brunn-Minkowski inequality.If p≥1p\ge 1, i=1,2,…,2n−1i=1,2,\ldots ,2n-1, and K∈Ko(Cn)K\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), there exists a positive Borel measure Sp,i(K,⋅){S}_{p,i}\left(K,\cdot )on S2n−1{S}^{2n-1}, such that the Lp{L}_{p}mixed quermassintegral Wp,i(K,L){W}_{p,i}\left(K,L)has the following integral representation (see [4]): (2.3)Wp,i(K,L)=12n∫S2n−1h(L,u)pdSp,i(K,u){W}_{p,i}\left(K,L)=\frac{1}{2n}\mathop{\int }\limits_{{S}^{2n-1}}h{\left(L,u)}^{p}{\rm{d}}{S}_{p,i}\left(K,u)for all L∈Ko(Cn)L\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}). It turns out that the measure Sp,i(K,⋅){S}_{p,i}\left(K,\cdot )on S2n−1{S}^{2n-1}is absolutely continuous with respect to Si(K,⋅){S}_{i}\left(K,\cdot )and has the Radon-Nikodym derivative dSp,i(K,⋅)dSi(K,⋅)=hK(⋅)1−p\frac{{\rm{d}}{S}_{p,i}\left(K,\cdot )}{{\rm{d}}{S}_{i}\left(K,\cdot )}={h}_{K}{\left(\cdot )}^{1-p}. Obviously, Sp,0(K,⋅)=Sp(K,⋅){S}_{p,0}\left(K,\cdot )={S}_{p}\left(K,\cdot ).In view of the Lp{L}_{p}Minkowski inequality in Rn{{\mathbb{R}}}^{n}by Lutwak [4], there is an Lp{L}_{p}Minkowski inequality about the Lp{L}_{p}mixed quermassintegrals in Cn{{\mathbb{C}}}^{n}. That is, if K,L∈Ko(Cn),p≥1K,L\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}),p\ge 1, and 0≤i<2n0\le i\lt 2n, then (2.4)Wp,i(K,L)2n−i≥Wi(K)2n−i−pWi(L)p{W}_{p,i}{\left(K,L)}^{2n-i}\ge {W}_{i}{\left(K)}^{2n-i-p}{W}_{i}{\left(L)}^{p}with equality for p=1p=1if and only if KKand LLare real homothetic, while for p>1p\gt 1if and only if KKand LLare real dilates. In particular, if i=0i=0in (2.4), then (2.5)Vp(K,L)2n≥V(K)2n−pV(L)p{V}_{p}{\left(K,L)}^{2n}\ge V{\left(K)}^{2n-p}V{\left(L)}^{p}with equality if and only if KKand LLare real dilates.2.3The dual Lp{L}_{p}mixed quermassintegralsFor K∈So(Cn)K\in {{\mathcal{S}}}_{o}\left({{\mathbb{C}}}^{n})and i=1,2,…,2n−1i=1,2,\ldots ,2n-1, the dual quermassintegrals W˜i(K){\widetilde{W}}_{i}\left(K)are defined by (see [30]) (2.6)W˜i(K)=12n∫S2n−1ρ(K,u)2n−idS(u),{\widetilde{W}}_{i}\left(K)=\frac{1}{2n}\mathop{\int }\limits_{{S}^{2n-1}}\rho {\left(K,u)}^{2n-i}{\rm{d}}S\left(u),where S(u)S\left(u)stands for the push forward with respect to ι−1{\iota }^{-1}of ℋ2n−1{{\mathcal{ {\mathcal H} }}}^{2n-1}on the (2n−1)\left(2n-1)-dimensional Euclidean unit sphere.If p≥1,K,L∈So(Cn)p\ge 1,K,L\in {{\mathcal{S}}}_{o}\left({{\mathbb{C}}}^{n}), and α,β≥0\alpha ,\beta \ge 0that are not both zero, the Lp{L}_{p}harmonic radial combination α⋅K+˜pβ⋅L\alpha \cdot K{\widetilde{+}}_{p}\beta \cdot Lis defined by ρα⋅K+˜pβ⋅L−p=αρK−p+βρL−p.{\rho }_{\alpha \cdot K{\widetilde{+}}_{p}\beta \cdot L}^{-p}=\alpha {\rho }_{K}^{-p}+\beta {\rho }_{L}^{-p}.For p≥1,i=1,2,…,2n−1p\ge 1,i=1,2,\ldots ,2n-1, and K,L∈So(Cn)K,L\in {{\mathcal{S}}}_{o}\left({{\mathbb{C}}}^{n}), the dual Lp{L}_{p}mixed quermassintegral W˜p,i(K,L){\widetilde{W}}_{p,i}\left(K,L)has the following integral representation (see [31]): (2.7)W˜−p,i(K,L)=12n∫S2n−1ρ(K,u)2n+p−iρ(L,u)−pdS(u).{\widetilde{W}}_{-p,i}\left(K,L)=\frac{1}{2n}\mathop{\int }\limits_{{S}^{2n-1}}\rho {\left(K,u)}^{2n+p-i}\rho {\left(L,u)}^{-p}{\rm{d}}S\left(u).The dual Lp{L}_{p}Minkowski inequality in Cn{{\mathbb{C}}}^{n}can be stated as follows: If p≥1p\ge 1, K,L∈So(Cn)K,L\in {{\mathcal{S}}}_{o}\left({{\mathbb{C}}}^{n}), and 0≤i<2n0\le i\lt 2n, then (2.8)W˜−p,i(K,L)2n−i≥W˜i(K)2n+p−iW˜i(L)−p{\widetilde{W}}_{-p,i}{\left(K,L)}^{2n-i}\ge {\widetilde{W}}_{i}{\left(K)}^{2n+p-i}{\widetilde{W}}_{i}{\left(L)}^{-p}with equality if and only if KKand LLare real dilates.2.4The general complex Lp{L}_{p}projection body and complex Lp{L}_{p}mixed projection bodySince the integral representations of the general complex Lp{L}_{p}projection body and complex Lp{L}_{p}mixed projection body need to be used in Section 3, we will present their integral representations in this part.First of all, by combining (1.3) and (1.4), we obtain the integral representation of the general complex Lp{L}_{p}projection body Πp,CλK{\Pi }_{p,C}^{\lambda }Kas (2.9)h(Πp,CλK,u)p=∫S2n−1∫Sn−1f1(λ)(ℛ[cu⋅v])+pdμp,C(c)dSp(K,v)+∫S2n−1∫Sn−1f2(λ)(ℛ[cu⋅v])−pdμp,C(c)dSp(K,v)h{\left({\Pi }_{p,C}^{\lambda }K,u)}^{p}=\mathop{\int }\limits_{{S}^{2n-1}}\mathop{\int }\limits_{{S}^{n-1}}{f}_{1}\left(\lambda ){\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{+}^{p}{\rm{d}}{\mu }_{p,C}\left(c){\rm{d}}{S}_{p}\left(K,v)+\mathop{\int }\limits_{{S}^{2n-1}}\mathop{\int }\limits_{{S}^{n-1}}{f}_{2}\left(\lambda ){\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{-}^{p}{\rm{d}}{\mu }_{p,C}\left(c){\rm{d}}{S}_{p}\left(K,v)for all u∈S2n−1u\in {S}^{2n-1}and every λ∈[−1,1]\lambda \in \left[-1,1].In order to give the integral representation of Πp,Cλ(K1,K2,…,K2n−1){\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}), we introduce the following definition.Definition 2.1For p≥0p\ge 0and K1,K2,…,K2n−1∈Ko(Cn){K}_{1},{K}_{2},\ldots ,{K}_{2n-1}\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), we define Borel measure Sp(K1,K2,…,K2n−1,⋅){S}_{p}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1},\cdot )on S2n−1{S}^{2n-1}as (2.10)Sp(K1,K2,…,K2n−1;ω)=∫ω(h(K1,u)h(K2,u)…h(K2n−1,u))1−p2n−1dS(K1,K2,…,K2n−1;u){S}_{p}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}\omega )=\mathop{\int }\limits_{\omega }{\left(h\left({K}_{1},u)h\left({K}_{2},u)\ldots h\left({K}_{2n-1},u))}^{\tfrac{1-p}{2n-1}}{\rm{d}}S\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}u)for each Borel ω⊂S2n−1\omega \subset {S}^{2n-1}.Let us mention that Sp(K1,K2,…,K2n−1;⋅){S}_{p}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}\cdot )on S2n−1{S}^{2n-1}is called the general complex Lp{L}_{p}mixed surface area measure of K1,K2,…,K2n−1{K}_{1},{K}_{2},\ldots ,{K}_{2n-1}and has the Radon-Nikodym derivative (2.11)dSp(K1,K2,…,K2n−1;⋅)dS(K1,K2,…,K2n−1;⋅)=(h(K1,⋅)h(K2,⋅)…h(K2n−1,⋅))1−p2n−1.\hspace{-26.85em}\frac{{\rm{d}}{S}_{p}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}\cdot )}{{\rm{d}}S\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}\cdot )}={\left(h\left({K}_{1},\cdot )h\left({K}_{2},\cdot )\ldots h\left({K}_{2n-1},\cdot ))}^{\tfrac{1-p}{2n-1}}.If K1=K2=⋯=K2n−1=K{K}_{1}={K}_{2}=\cdots ={K}_{2n-1}=K, then Sp(K,K,…,K,⋅)=Sp(K,⋅){S}_{p}\left(K,K,\ldots ,K,\cdot )={S}_{p}\left(K,\cdot ). In particular, if K1,K2,…,K2n−1−i∈Ko(Cn),K2n−i=⋯=K2n−1=B{K}_{1},{K}_{2},\ldots ,{K}_{2n-1-i}\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}),{K}_{2n-i}=\cdots ={K}_{2n-1}=B, then for i=0,1,…,2n−2i=0,1,\ldots ,2n-2, we obtain (2.12)dSp,i(K1,K2,…,K2n−1−i;⋅)dSi(K1,K2,…,K2n−1−i;⋅)=(h(K1,⋅)h(K2,⋅)…h(K2n−1−i,⋅))1−p2n−1−i.\frac{{\rm{d}}{S}_{p,i}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1-i};\hspace{0.33em}\cdot )}{{\rm{d}}{S}_{i}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1-i};\hspace{0.33em}\cdot )}={\left(h\left({K}_{1},\cdot )h\left({K}_{2},\cdot )\ldots h\left({K}_{2n-1-i},\cdot ))}^{\tfrac{1-p}{2n-1-i}}.Next, with respect to (1.7) and (2.10), we have the following integral representation of the general complex Lp{L}_{p}mixed projection body Πp,Cλ(K1,K2,…,K2n−1){\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})as (2.13)h(Πp,Cλ(K1,K2,…,K2n−1),u)p=∫S2n−1∫Sn−1f1(λ)(ℛ[cu⋅v])+pdSp(K1,K2,…,K2n−1;v)+∫S2n−1∫Sn−1f2(λ)(ℛ[cu⋅v])+pdSp(K1,K2,…,K2n−1;v)\begin{array}{rcl}h{\left({\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),u)}^{p}& =& \mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{n-1}}{f}_{1}\left(\lambda ){\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{+}^{p}{\rm{d}}{S}_{p}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}v)\\ & & +\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{n-1}}{f}_{2}\left(\lambda ){\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{+}^{p}{\rm{d}}{S}_{p}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}v)\end{array}for all u,v∈S2n−1u,v\in {S}^{2n-1}and every λ∈[−1,1]\lambda \in \left[-1,1].3Proofs of main resultsIn this section, we give the proofs of Theorems 1.1–1.4. First, the proof of Theorem 1.1 needs the following Lemma 3.1.Lemma 3.1Let K∈Ko(Cn)K\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n})and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})be an asymmetric Lp{L}_{p}zonoid. If p≥1p\ge 1and 0≤i<2n0\le i\lt 2n, then for all Q∈Ko(Cn)Q\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), we have(3.1)Wp,i(Q,Πp,CλK)=Vp(K,Πp,i,C¯λQ).{W}_{p,i}\left(Q,{\Pi }_{p,C}^{\lambda }K)={V}_{p}\left(K,{\Pi }_{p,i,\bar{C}}^{\lambda }Q).ProofFrom (2.3), (1.4), and the conjugate linear of Hermitian inner product, we know that for all Q∈Ko(Cn)Q\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), Wp,i(Q,Πp,CλK)=12n∫S2n−1h(Πp,CλK,u)pdSp,i(Q,u)=12n∫S2n−1∫S2n−1∫Sn−1f1(λ)(ℛ[cu⋅v])+pdμp,C(c)dSp(K,v)dSp,i(Q,u)+12n∫S2n−1∫S2n−1∫Sn−1f2(λ)(ℛ[cu⋅v])−pdμp,C(c)dSp(K,v)dSp,i(Q,u)=12n∫S2n−1∫S2n−1∫Sn−1f1(λ)(ℛ[u⋅c¯v])+pdμp,C(c)dSp,i(Q,u)Sp(K,v)+12n∫S2n−1∫S2n−1∫Sn−1f2(λ)(ℛ[u⋅c¯v])−pdμp,C(c)dSp,i(Q,u)Sp(K,v)=12n∫S2n−1∫S2n−1f1(λ)h(C¯u,v)pdSp,i(Q,u)dSp(K,v)+12n∫S2n−1∫S2n−1f2(λ)h(−C¯u,v)pdSp,i(Q,u)dSp(K,v)=12n∫S2n−1h(Πp,i,C¯λQ,v)pdSp(K,v)=Vp(K,Πp,i,C¯λQ),\begin{array}{rcl}{W}_{p,i}\left(Q,{\Pi }_{p,C}^{\lambda }K)& =& \frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}h{\left({\Pi }_{p,C}^{\lambda }K,u)}^{p}{\rm{d}}{S}_{p,i}\left(Q,u)\\ & =& \frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{n-1}}{f}_{1}\left(\lambda ){\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{+}^{p}{\rm{d}}{\mu }_{p,C}\left(c){\rm{d}}{S}_{p}\left(K,v){\rm{d}}{S}_{p,i}\left(Q,u)\\ & & +\frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{n-1}}{f}_{2}\left(\lambda ){\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{-}^{p}{\rm{d}}{\mu }_{p,C}\left(c){\rm{d}}{S}_{p}\left(K,v){\rm{d}}{S}_{p,i}\left(Q,u)\\ & =& \frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{n-1}}{f}_{1}\left(\lambda ){\left({\mathcal{ {\mathcal R} }}\left[u\cdot \bar{c}v])}_{+}^{p}{\rm{d}}{\mu }_{p,C}\left(c){\rm{d}}{S}_{p,i}\left(Q,u){S}_{p}\left(K,v)\\ & & +\frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{n-1}}{f}_{2}\left(\lambda ){\left({\mathcal{ {\mathcal R} }}\left[u\cdot \bar{c}v])}_{-}^{p}{\rm{d}}{\mu }_{p,C}\left(c){\rm{d}}{S}_{p,i}\left(Q,u){S}_{p}\left(K,v)\\ & =& \frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}{f}_{1}\left(\lambda )h{\left(\bar{C}u,v)}^{p}{\rm{d}}{S}_{p,i}\left(Q,u){\rm{d}}{S}_{p}\left(K,v)\\ & & +\frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}{f}_{2}\left(\lambda )h{\left(-\bar{C}u,v)}^{p}{\rm{d}}{S}_{p,i}\left(Q,u){\rm{d}}{S}_{p}\left(K,v)\\ & =& \frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}h{\left({\Pi }_{p,i,\bar{C}}^{\lambda }Q,v)}^{p}{\rm{d}}{S}_{p}\left(K,v)\\ & =& {V}_{p}\left(K,{\Pi }_{p,i,\bar{C}}^{\lambda }Q),\end{array}where C¯\bar{C}is the conjugate of CCand then we conclude the proof.□Now, we are in a position to prove Theorem 1.1.Proof of Theorem 1.1Since K,L∈Kos(Cn)K,L\in {{\mathcal{K}}}_{os}\left({{\mathbb{C}}}^{n}), then for N∈Kos(Cn)N\in {{\mathcal{K}}}_{os}\left({{\mathbb{C}}}^{n})and by (1.8), we have (3.2)Vp(K#pL,N)=12n∫S2n−1h(N,u)pdSp(K#pL)=Vp(K,N)+Vp(L,N).{V}_{p}\left(K\hspace{-0.3em}{\text{\#}}_{p}L,N)=\frac{1}{2n}\mathop{\int }\limits_{{S}^{2n-1}}h{\left(N,u)}^{p}{\rm{d}}{S}_{p}\left(K\hspace{-0.3em}{\text{\#}}_{p}L)={V}_{p}\left(K,N)+{V}_{p}\left(L,N).According to Lemma 3.1, for all Q∈Kos(Cn)Q\in {{\mathcal{K}}}_{os}\left({{\mathbb{C}}}^{n}), one deduces Wp,i(Q,Πp,Cλ(K#pL))=Vp(K#pL,Πp,C¯λQ)=Vp(K,Πp,i,C¯λQ)+Vp(L,Πp,i,C¯λQ)=Wp,i(Q,Πp,CλK)+Wp,i(Q,Πp,CλL).\hspace{-27.5em}\begin{array}{rcl}{W}_{p,i}\left(Q,{\Pi }_{p,C}^{\lambda }\left(K\hspace{-0.3em}{\text{\#}}_{p}L))& =& {V}_{p}\left(K\hspace{-0.3em}{\text{\#}}_{p}L,{\Pi }_{p,\bar{C}}^{\lambda }Q)\\ & =& {V}_{p}\left(K,{\Pi }_{p,i,\bar{C}}^{\lambda }Q)+{V}_{p}\left(L,{\Pi }_{p,i,\bar{C}}^{\lambda }Q)\\ & =& {W}_{p,i}\left(Q,{\Pi }_{p,C}^{\lambda }K)+{W}_{p,i}\left(Q,{\Pi }_{p,C}^{\lambda }L).\end{array}By (2.4), we have (3.3)Wp,i(Q,Πp,Cλ(K#pL))≥Wi(Q)2n−p−i2n−iWi(Πp,CλK)p2n−i+Wi(Πp,CλL)p2n−i\hspace{-27.35em}{W}_{p,i}\left(Q,{\Pi }_{p,C}^{\lambda }\left(K\hspace{-0.3em}{\text{\#}}_{p}L))\ge {W}_{i}{\left(Q)}^{\tfrac{2n-p-i}{2n-i}}\left[{W}_{i}{\left({\Pi }_{p,C}^{\lambda }K)}^{\tfrac{p}{2n-i}}+{W}_{i}{\left({\Pi }_{p,C}^{\lambda }L)}^{\tfrac{p}{2n-i}}\right]with equality if and only if Q,KQ,K, and LLare real dilates.Taking Q=Πp,Cλ(K#pL)Q={\Pi }_{p,C}^{\lambda }\left(K\hspace{-0.3em}{\text{\#}}_{p}L)in (3.3), we obtain (3.4)Wi(Πp,Cλ(K#pL))p2n−i≥Wi(Πp,CλK)p2n−i+Wi(Πp,CλL)p2n−i\hspace{-27.35em}{W}_{i}{\left({\Pi }_{p,C}^{\lambda }\left(K{\text{\#}}_{p}L))}^{\tfrac{p}{2n-i}}\ge {W}_{i}{\left({\Pi }_{p,C}^{\lambda }K)}^{\tfrac{p}{2n-i}}+{W}_{i}{\left({\Pi }_{p,C}^{\lambda }L)}^{\tfrac{p}{2n-i}}with equality if and only KKand LLare real dilates.□The following lemma provides a connection of Πp,Cλ,∗{\Pi }_{p,C}^{\lambda ,\ast }and Mp,i,C¯λ{M}_{p,i,\bar{C}}^{\lambda }in terms of the dual Lp{L}_{p}mixed quermassintergral and their mixed volume. We need the lemma to prove Theorem 1.2.Lemma 3.2Let K∈Ko(Cn)K\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n})and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})be an asymmetric Lp{L}_{p}zonoid. If p≥1p\ge 1and 0≤i<2n0\le i\lt 2n, then for all Q∈So(Cn)Q\in {{\mathcal{S}}}_{o}\left({{\mathbb{C}}}^{n}), we have(3.5)W˜−p,i(Q,Πp,Cλ,∗K)=2n+p2Vp(K,Mp,i,C¯λQ),{\widetilde{W}}_{-p,i}\left(Q,{\Pi }_{p,C}^{\lambda ,\ast }K)=\frac{2n+p}{2}{V}_{p}\left(K,{M}_{p,i,\bar{C}}^{\lambda }Q),where Mp,CλQ{M}_{p,C}^{\lambda }Qis the general complex Lp{L}_{p}moment body [24]. If K2n−i=⋯=K2n−1=B{K}_{2n-i}=\cdots ={K}_{2n-1}=B, we write Mp,i,Cλ(Q,B){M}_{p,i,C}^{\lambda }\left(Q,B)as Mp,i,CλQ{M}_{p,i,C}^{\lambda }Q.ProofBy (2.7), (2.9), and the conjugate linear of Hermitian inner product, we have Vp(K,Mp,i,C¯λQ)=12n∫S2n−1h(Mp,i,C¯λQ,u)pdSp(K,u)=22n(2n+p)∫S2n−1∫S2n−1f1(λ)h(C¯u,v)pρ(Q,v)2n+p−idS(v)dSp(K,u)+22n(2n+p)∫S2n−1∫S2n−1f2(λ)h(−C¯u,v)pρ(Q,v)2n+p−idS(v)dSp(K,u)=22n(2n+p)∫S2n−1∫S2n−1f1(λ)h(Cu,v)pρ(Q,v)2n+p−idSp(K,u)dS(v)+22n(2n+p)∫S2n−1∫S2n−1f2(λ)h(−Cu,v)pρ(Q,v)2n+p−idS(v)dSp(K,u)=22n(2n+p)∫S2n−1h(Πp,CλK,v)pρ(Q,v)2n+p−idS(v)=22n(2n+p)∫S2n−1ρ(Πp,Cλ,∗K,v)−pρ(Q,v)2n+p−idS(v)=22n+pW˜−p,i(Q,Πp,Cλ,∗K),\begin{array}{rcl}{V}_{p}\left(K,{M}_{p,i,\bar{C}}^{\lambda }Q)& =& \frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}h{\left({M}_{p,i,\bar{C}}^{\lambda }Q,u)}^{p}{\rm{d}}{S}_{p}\left(K,u)\\ & =& \frac{2}{2n\left(2n+p)}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}{f}_{1}\left(\lambda )h{\left(\bar{C}u,v)}^{p}\rho {\left(Q,v)}^{2n+p-i}{\rm{d}}S\left(v){\rm{d}}{S}_{p}\left(K,u)\\ & & +\frac{2}{2n\left(2n+p)}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}{f}_{2}\left(\lambda )h{\left(-\bar{C}u,v)}^{p}\rho {\left(Q,v)}^{2n+p-i}{\rm{d}}S\left(v){\rm{d}}{S}_{p}\left(K,u)\\ & =& \frac{2}{2n\left(2n+p)}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}{f}_{1}\left(\lambda )h{\left(Cu,v)}^{p}\rho {\left(Q,v)}^{2n+p-i}{\rm{d}}{S}_{p}\left(K,u){\rm{d}}S\left(v)\\ & & +\frac{2}{2n\left(2n+p)}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}{f}_{2}\left(\lambda )h{\left(-Cu,v)}^{p}\rho {\left(Q,v)}^{2n+p-i}{\rm{d}}S\left(v){\rm{d}}{S}_{p}\left(K,u)\\ & =& \frac{2}{2n\left(2n+p)}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}h{\left({\Pi }_{p,C}^{\lambda }K,v)}^{p}\rho {\left(Q,v)}^{2n+p-i}{\rm{d}}S\left(v)\\ & =& \frac{2}{2n\left(2n+p)}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\rho {\left({\Pi }_{p,C}^{\lambda ,\ast }K,v)}^{-p}\rho {\left(Q,v)}^{2n+p-i}{\rm{d}}S\left(v)\\ & =& \frac{2}{2n+p}{\widetilde{W}}_{-p,i}\left(Q,{\Pi }_{p,C}^{\lambda ,\ast }K),\end{array}which ends the proof of Lemma 3.2.□Proof of Theorem 1.2From (2.8), (3.2), and Lemma 3.2, we have for all Q∈So(Cn)Q\in {{\mathcal{S}}}_{o}\left({{\mathbb{C}}}^{n})(3.6)W˜−p,i(Q,Πp,Cλ,∗(K#pL))=2n+p2Vp(K#pL,Mp,i,C¯λQ)=2n+p2[Vp(K,Mp,i,C¯λQ)+Vp(L,Mp,i,C¯λQ)]=W˜−p,i(Q,Πp,Cλ,∗K)+W˜−p,i(Q,Πp,Cλ,∗L)≥W˜i(Q)2n+p−i2n−iW˜i(Πp,Cλ,∗K)−p2n−i+W˜i(Πp,Cλ,∗L)−p2n−i\hspace{-34.75em}\begin{array}{rcl}{\widetilde{W}}_{-p,i}\left(Q,{\Pi }_{p,C}^{\lambda ,\ast }\left(K\hspace{-0.3em}{\text{\#}}_{p}L))& =& \frac{2n+p}{2}{V}_{p}\left(K\hspace{-0.3em}{\text{\#}}_{p}L,{M}_{p,i,\bar{C}}^{\lambda }Q)\\ & =& \frac{2n+p}{2}{[}{V}_{p}\left(K,{M}_{p,i,\bar{C}}^{\lambda }Q)+{V}_{p}\left(L,{M}_{p,i,\bar{C}}^{\lambda }Q)]\\ & =& {\widetilde{W}}_{-p,i}\left(Q,{\Pi }_{p,C}^{\lambda ,\ast }K)+{\widetilde{W}}_{-p,i}\left(Q,{\Pi }_{p,C}^{\lambda ,\ast }L)\\ & \ge & {\widetilde{W}}_{i}{\left(Q)}^{\tfrac{2n+p-i}{2n-i}}\left[{\widetilde{W}}_{i}{\left({\Pi }_{p,C}^{\lambda ,\ast }K)}^{-\tfrac{p}{2n-i}}+{\widetilde{W}}_{i}{\left({\Pi }_{p,C}^{\lambda ,\ast }L)}^{-\tfrac{p}{2n-i}}\right]\end{array}with equality if and only if Q,KQ,K, and LLare real dilates.Taking Q=Πp,Cλ,∗(K#pL)Q={\Pi }_{p,C}^{\lambda ,\ast }\left(K\hspace{-0.3em}{\text{\#}}_{p}L)in (3.6), it yields that (3.7)W˜i(Πp,Cλ,∗(K#pL))−p2n−i≥W˜i(Πp,Cλ,∗K)−p2n−i+W˜i(Πp,Cλ,∗L)−p2n−i{\widetilde{W}}_{i}{\left({\Pi }_{p,C}^{\lambda ,\ast }\left(K{\text{\#}}_{p}L))}^{-\tfrac{p}{2n-i}}\ge {\widetilde{W}}_{i}{\left({\Pi }_{p,C}^{\lambda ,\ast }K)}^{-\tfrac{p}{2n-i}}+{\widetilde{W}}_{i}{\left({\Pi }_{p,C}^{\lambda ,\ast }L)}^{-\tfrac{p}{2n-i}}with equality if and only if KKand LLare real dilates. Finally, let i=0i=0in (3.7), we conclude the proof of Theorem 1.2.□From now on, we pay attention to prove Theorems 1.3 and 1.4.Proof of Theorem 1.3From (1.3), (2.11), and the Hölder’s integral inequality [32], one has (3.8)h(Πp,C+(K1,K2,…,K2n−1),u)p=∫S2n−1h(C⋅u,v)pdSp(K1,K2,…,K2n−1;v)=∫S2n−1h(C⋅u,v)p(h(K1,u)⋯h(K2n−1,u))1−p2n−1dS(K1,K2,…,K2n−1;v)≥∫S2n−1h(C⋅u,v)dS(K1,K2,…,K2n−1;v)p(2n)1−p∏j=12n−1V(K1,K2,…,K2n−1,Kj)1−p2n−1=h(ΠC(K1,K2,…,K2n−1),u)p(2n)1−p∏j=12n−1V(K1,K2,…,K2n−1,Kj)1−p2n−1.\begin{array}{rcl}h{\left({\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),u)}^{p}& =& \mathop{\displaystyle \int }\limits_{{S}^{2n-1}}h{\left(C\cdot u,v)}^{p}{\rm{d}}{S}_{p}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}v)\\ & =& \mathop{\displaystyle \int }\limits_{{S}^{2n-1}}h{\left(C\cdot u,v)}^{p}{\left(h\left({K}_{1},u)\cdots h\left({K}_{2n-1},u))}^{\tfrac{1-p}{2n-1}}{\rm{d}}S\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}v)\\ & \ge & {\left(\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}h\left(C\cdot u,v){\rm{d}}S\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};v)\right)}^{p}{\left(2n)}^{1-p}\mathop{\displaystyle \prod }\limits_{j=1}^{2n-1}\\ & & V{\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}\\ & =& h{\left({\Pi }_{C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),u)}^{p}{\left(2n)}^{1-p}\mathop{\displaystyle \prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}.\end{array}According to the equality condition of Hölder’s integral inequality, the equality holds if and only if Kj{K}_{j}and C⋅uC\cdot uare dilates.For each Q∈Ko(Cn)Q\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), integrating both sides of (3.8) for dSp,i(Q,u){\rm{d}}{S}_{p,i}\left(Q,u)in u∈S2n−1u\in {S}^{2n-1}, we obtain Wp,i(Q,Πp,C+(K1,K2,…,K2n−1))≥(2n)1−p∏j=12n−1V(K1,K2,…,K2n−1,Kj)1−p2n−1Wp,i(Q,ΠC(K1,K2,…,K2n−1)).{W}_{p,i}\left(Q,{\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}{W}_{p,i}\left(Q,{\Pi }_{C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})).Taking Q=Πp,C+(K1,K2,…,K2n−1)Q={\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})and by (2.4), we have (3.9)Wi(Πp,C+(K1,K2,…,K2n−1))p2n−i≥(2n)1−p∏j=12n−1V(K1,K2,…,K2n−1,Kj)1−p2n−1Wi(ΠC(K1,K2,…,K2n−1))p2n−i\hspace{-39.85em}{W}_{i}{\left({\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n-i}}\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}{W}_{i}{\left({\Pi }_{C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n-i}}with equality if and only if Πp,C+(K1,K2,…,K2n−1),ΠC(K1,K2,…,K2n−1){\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),{\Pi }_{C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})are real dilates. But the equality condition in inequality (3.8) reveals that this happens precisely if Kj{K}_{j}and C⋅uC\cdot uare dilates.By the extension of the Brunn-Minkowski inequality (2.2), we obtain (3.10)Wi(Πp,Cλ(K1,K2,…,K2n−1))p2n−i≥Wi(Πp,C±(K1,K2,…,K2n−1))p2n−i{W}_{i}{\left({\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n-i}}\ge {W}_{i}{\left({\Pi }_{p,C}^{\pm }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n-i}}with equality if and only if Πp,C+(K1,K2,…,K2n−1),Πp,C−(K1,K2,…,K2n−1){\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),{\Pi }_{p,C}^{-}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})are real dilates which is only possible if Πp,C−(K1,K2,…,K2n−1)=Πp,C+(K1,K2,…,K2n−1){\Pi }_{p,C}^{-}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})={\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}). It means that if Πp,C−(K1,K2,…,K2n−1)≠Πp,C+(K1,K2,…,K2n−1){\Pi }_{p,C}^{-}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})\ne {\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}), the inequality is strict unless λ=−1,1\lambda =-1,1, or 0.Combining (3.9) and (3.10), we obtain (3.11)Wi(Πp,Cλ(K1,K2,…,K2n−1))p2n−i≥(2n)1−p∏j=12n−1V(K1,K2,…,K2n−1,Kj)1−p2n−1Wi(ΠC(K1,K2,…,K2n−1))p2n−i.{W}_{i}{\left({\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n-i}}\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}{W}_{i}{\left({\Pi }_{C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n-i}}.If p=1p=1, the Aleksandrov-Fenchel-type inequality (see [22]) is V(ΠC(K1,K2,…,K2n−1))r≥∏j=1rV(ΠC(Kj[r],Kr+1,…,K2n−1)).V{\left({\Pi }_{C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{r}\ge \mathop{\prod }\limits_{j=1}^{r}V\left({\Pi }_{C}\left({K}_{j}\left[r],{K}_{r+1},\ldots ,{K}_{2n-1})).After that, let i=0i=0in (3.11) and combine with the case of p=1p=1, we obtain (3.12)V(Πp,Cλ(K1,K2,…,K2n−1))p2n≥(2n)1−p∏j=12n−1V(K1,…,K2n−1,Kj)1−p2n−1∏j=1rV(ΠC(Kj[r],Kr+1,…,K2n−1))p2nr.V{\left({\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n}}\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}\mathop{\prod }\limits_{j=1}^{r}V{\left({\Pi }_{C}\left({K}_{j}\left[r],{K}_{r+1},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2nr}}.Let us turn toward the equality condition. If Kj{K}_{j}is an ellipsoid centered at the origin, then Kj=ϕB{K}_{j}=\phi Bfor ϕ∈GL(n,C)\phi \in GL\left(n,{\mathbb{C}}). From Πp,Cλ(ϕB)=∣detϕ∣2pϕ−∗Πp,CλB{\Pi }_{p,C}^{\lambda }\left(\phi B)=| {\rm{\det }}\hspace{0.33em}\phi \hspace{-0.25em}{| }^{\tfrac{2}{p}}{\phi }^{-\ast }{\Pi }_{p,C}^{\lambda }Band the fact that Πp,CλB=aB,ΠCB=bBa,b>0{\Pi }_{p,C}^{\lambda }B=aB,{\Pi }_{C}B=bB\hspace{0.33em}a,b\gt 0(see [20,24]), we have (3.13)V(Πp,Cλ(K1,K2,…,K2n−1))p2n=V(Πp,Cλ(ϕB))p2n=V(∣detϕ∣2pϕ−∗Πp,CλB)p2n=∣detϕ∣2V(ϕ−∗Πp,CλB)p2n=∣detϕ∣2∣detϕ−∗∣pnV(aB)p2n=∣detϕ∣2∣detϕ−∗∣pnapV(B)p2n.\hspace{-29.9em}\begin{array}{rcl}V{\left({\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n}}& =& V{\left({\Pi }_{p,C}^{\lambda }\left(\phi B))}^{\tfrac{p}{2n}}\\ & =& V{\left(| {\rm{\det }}\phi {| }^{\frac{2}{p}}{\phi }^{-\ast }{\Pi }_{p,C}^{\lambda }B)}^{\tfrac{p}{2n}}\\ & =& | {\rm{\det }}\hspace{0.33em}\phi \hspace{-0.25em}{| }^{2}V{\left({\phi }^{-\ast }{\Pi }_{p,C}^{\lambda }B)}^{\tfrac{p}{2n}}\\ & =& | {\rm{\det }}\hspace{0.33em}\phi \hspace{-0.25em}{| }^{2}| {\rm{\det }}\hspace{0.33em}{\phi }^{-\ast }\hspace{-0.25em}{| }^{\tfrac{p}{n}}V{\left(aB)}^{\tfrac{p}{2n}}\\ & =& | {\rm{\det }}\hspace{0.33em}\phi \hspace{-0.25em}{| }^{2}| {\rm{\det }}\hspace{0.33em}{\phi }^{-\ast }\hspace{-0.25em}{| }^{\tfrac{p}{n}}{a}^{p}V{\left(B)}^{\tfrac{p}{2n}}.\end{array}Similarly, we also obtain (3.14)V(ΠC(K1,K2,…,K2n−1))p2n=V(ΠC(ϕB))p2n=∣detϕ∣2p∣detϕ−∗∣pnbpV(B)p2n,V{\left({\Pi }_{C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n}}=V{\left({\Pi }_{C}\left(\phi B))}^{\tfrac{p}{2n}}=| {\rm{\det }}\hspace{0.33em}\phi \hspace{-0.25em}{| }^{2p}| {\rm{\det }}\hspace{0.33em}{\phi }^{-\ast }\hspace{-0.25em}{| }^{\tfrac{p}{n}}{b}^{p}V{\left(B)}^{\tfrac{p}{2n}},(3.15)(2n)1−p∏j=12n−1V(K1,…,K2n−1,Kj)1−p2n−1=(2n)1−pV(ϕB)1−p=(2n)1−p∣detϕ∣2(1−p)V(B)1−p=c∣detϕ∣2(1−p),{\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}={\left(2n)}^{1-p}V{\left(\phi B)}^{1-p}={\left(2n)}^{1-p}| {\rm{\det }}\hspace{0.33em}\phi \hspace{-0.25em}{| }^{2\left(1-p)}V{\left(B)}^{1-p}=c| {\rm{\det }}\hspace{0.33em}\phi \hspace{-0.25em}{| }^{2\left(1-p)},where c=∫S2n−1h(B,u)dS(B,u)1−pc={\left({\int }_{{S}^{2n-1}}h\left(B,u){\rm{d}}S\left(B,u)\right)}^{1-p}is a constant.From (3.14) and (3.15), we have (3.16)(2n)1−p∏j=12n−1V(K1,…,K2n−1,Kj)1−p2n−1∏j=1rV(ΠC(Kj[r],Kr+1,…,K2n−1))p2nr=c∣detϕ∣2∣detϕ−∗∣pnbpV(B)p2n.{\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}\mathop{\prod }\limits_{j=1}^{r}V{\left({\Pi }_{C}\left({K}_{j}\left[r],{K}_{r+1},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2nr}}=c| {\rm{\det }}\hspace{0.33em}\phi \hspace{-0.25em}{| }^{2}| {\rm{\det }}\hspace{0.33em}{\phi }^{-\ast }\hspace{-0.25em}{| }^{\tfrac{p}{n}}{b}^{p}V{\left(B)}^{\tfrac{p}{2n}}.Comparing equations (3.13) and (3.16), we know that a=c1pba={c}^{\tfrac{1}{p}}bis possible, which means that if Kj{K}_{j}is an ellipsoid centered at the origin, the equality holds in (3.12).If Kj{K}_{j}is an Hermitian ellipsoid, there exists a positive Hermitian matrix ϕ∈GL(n,C)\phi \in GL\left(n,{\mathbb{C}})and a vector t∈Cnt\in {{\mathbb{C}}}^{n}such that Kj=ϕB+t{K}_{j}=\phi B+t. The definition of Πp,Cλ{\Pi }_{p,C}^{\lambda }reveals that Πp,Cλ{\Pi }_{p,C}^{\lambda }is translation invariant. Hence, if Kj{K}_{j}is an Hermitian ellipsoid, equality also holds in (3.12).□The case of λ=0\lambda =0of Theorem 1.3 is the following Aleksandrov-Fenchel-type inequality for the complex Lp{L}_{p}mixed projection bodies.Corollary 3.1If p>1p\gt 1, K1,K2,…,K2n−1∈Ko(Cn){K}_{1},{K}_{2},\ldots ,{K}_{2n-1}\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})is an asymmetric Lp{L}_{p}zonoid, thenV(Πp,C(K1,K2,…,K2n−1))p2n≥(2n)1−p∏j=12n−1V(K1,…,K2n−1,Kj)1−p2n−1∏j=1rV(ΠC(Kj[r],Kr+1,…,K2n−1))p2nr.V{\left({\Pi }_{p,C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n}}\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}\mathop{\prod }\limits_{j=1}^{r}V{\left({\Pi }_{C}\left({K}_{j}\left[r],{K}_{r+1},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2nr}}.For p=1p=1, the inequality is Aleksandrov-Fenchel-type inequality for mixed projection bodies [22].In particular, the case of C=[0,1]C=\left[0,1]and λ=0\lambda =0of Theorem 1.3 is Aleksandrov-Fenchel-type inequality for the general Lp{L}_{p}mixed projection bodies [27]. We are now ready to prove Theorem 1.4. Note that Theorem 1.3 reduces to Theorem 1.4 by setting K2n−i=⋯=K2n−1=B{K}_{2n-i}=\cdots ={K}_{2n-1}=B.Proof of Theorem 1.4Recall that M≔(K1,K2,…,K2n−1−i){\bf{M}}:= \left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1-i}). From (2.12), (2.13), and the Hölder’s integral inequality, we have (3.17)h(Πp,i,CλM,u)p≥(2n)1−p∏j=12n−1−iWi(M,Kj)1−p2n−1−ih(Πi,CM,u)p.h{\left({\Pi }_{p,i,C}^{\lambda }{\bf{M}},u)}^{p}\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1-i}{W}_{i}{\left({\bf{M}},{K}_{j})}^{\tfrac{1-p}{2n-1-i}}h{\left({\Pi }_{i,C}{\bf{M}},u)}^{p}.For all Q∈Ko(Cn)Q\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), we integrate both sides of (3.17) for dSp(Q,u){\rm{d}}{S}_{p}\left(Q,u)and obtain Vp(Q,Πp,i,CλM)≥(2n)1−pVp(Q,Πi,CM)∏j=12n−1−iWi(M,Kj)1−p2n−1−i.{V}_{p}\left(Q,{\Pi }_{p,i,C}^{\lambda }{\bf{M}})\ge {\left(2n)}^{1-p}{V}_{p}\left(Q,{\Pi }_{i,C}{\bf{M}})\mathop{\prod }\limits_{j=1}^{2n-1-i}{W}_{i}{\left({\bf{M}},{K}_{j})}^{\tfrac{1-p}{2n-1-i}}.Taking Q=Πp,i,CλMQ={\Pi }_{p,i,C}^{\lambda }{\bf{M}}and using (2.5), we obtain V(Πp,i,CλM)p2n≥(2n)1−p∏j=1rV(Πi,C(Kj[r],Kr+1,…,K2n−1−i))p2nr∏j=12n−1−iWi(M,Kj)1−p2n−1−i.V{\left({\Pi }_{p,i,C}^{\lambda }{\bf{M}})}^{\tfrac{p}{2n}}\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{r}V{\left({\Pi }_{i,C}\left({K}_{j}\left[r],{K}_{r+1},\ldots ,{K}_{2n-1-i}))}^{\tfrac{p}{2nr}}\mathop{\prod }\limits_{j=1}^{2n-1-i}{W}_{i}{\left({\bf{M}},{K}_{j})}^{\tfrac{1-p}{2n-1-i}}.If Kj{K}_{j}is an ellipsoid centered at the origin or an Hermitian ellipsoid, then the equality holds.□ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

General complex Lp projection bodies and complex Lp mixed projection bodies

Open Mathematics , Volume 20 (1): 12 – Jan 1, 2022

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de Gruyter
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© 2022 Manli Cheng et al., published by De Gruyter
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2391-5455
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2391-5455
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10.1515/math-2022-0027
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Abstract

1IntroductionThe classical Brunn-Minkowski theory appeared at the turn of the nineteenth into the twentieth century. One of the core concepts that Minkowski introduced within the Brunn-Minkowski theory is the projection body. There are important inequalities involving the volume of the projection body and its polar, such as Petty projection inequality [1] and Zhang projection inequality [2].In the early 1960s, Firey [3] introduced the Firey-Minkowski Lp{L}_{p}-addition of a convex body. In the mid-1990s, it was shown in [4] and [5] that a study of the volume of the Firey-Minkowski Lp{L}_{p}combinations leads to the Lp{L}_{p}Brunn-Minkowski theory. This theory has expanded rapidly. An early achievement of the Lp{L}_{p}Brunn-Minkowski theory was the discovery of Lp{L}_{p}projection body, introduced by Lutwak et al. [6]. Since then, Ludwig [7,8] extended the projection body to an entire class that can be called the general Lp{L}_{p}projection body.A mixed projection body was introduced in the classic volume of Bonnesen-Fenchel [9]. In [10] and [11], Lutwak considered the volume of the mixed projection body and established the classical mixed volume inequalities, such as Aleksandrov-Fenchel inequalities and Brunn-Minkowski inequalities.Let us mention that the projection bodies described above are all real. The theory of the real projection body continues to be a very active field now. For additional information and some results on real projection body see, e.g., [8,12, 13,14,15, 16,17]. However, some classical concepts of convex geometry in real vector space were extended to complex cases, such as complex difference body [18], complex intersection body [19], complex centroid body [20,21], and complex projection body [22,23, 24,25].In this paper, we mainly study the projection body in complex vector space. Let K(Cn){\mathcal{K}}\left({{\mathbb{C}}}^{n})be the set of convex body (nonempty compact convex subsets) in complex vector space Cn{{\mathbb{C}}}^{n}. For the set of the convex body containing the origin in their interiors and the set of an origin-symmetric convex body in Cn{{\mathbb{C}}}^{n}, we write Ko(Cn){{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n})and Kos(Cn){{\mathcal{K}}}_{os}\left({{\mathbb{C}}}^{n}), respectively. Let V(K)V\left(K)denote the complex volume of KK, BBthe complex unit ball, and S2n−1{S}^{2n-1}the complex unit sphere.In 2011, the complex Lp{L}_{p}projection body ΠCK{\Pi }_{C}Kwas defined by Abardia and Bernig [22]: For K∈K(Cn)K\in {\mathcal{K}}\left({{\mathbb{C}}}^{n})and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}}), (1.1)h(ΠCK,u)=nV1(K,C⋅u)=12∫S2n−1h(C⋅u,v)dSK(v)h\left({\Pi }_{C}K,u)=n{V}_{1}\left(K,C\cdot u)=\frac{1}{2}\mathop{\int }\limits_{{S}^{2n-1}}h\left(C\cdot u,v){\rm{d}}{S}_{K}\left(v)\hspace{1.6em}for every u∈S2n−1u\in {S}^{2n-1}, where C⋅u≔{cu:c∈C}C\cdot u:= \{cu:c\in C\}. They also defined the mixed complex projection body as (1.2)h(ΠC(K1,K2,…,K2n−1),u)=nV(K1,K2,…,K2n−1,C⋅u).h\left({\Pi }_{C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),u)=nV\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1},C\cdot u).Very recently, the concept of asymmetric complex Lp{L}_{p}projection body Πp,C+K{\Pi }_{p,C}^{+}Kwas introduced in [24]. First, a convex body C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})is called an asymmetric Lp{L}_{p}zonoid if there exists a finite even Borel measure μp,C(v){\mu }_{p,C}\left(v)on Sn−1{S}^{n-1}such that hC(u)p=∫Sn−1(ℛ[cu⋅v])+pdμp,C(c).{h}_{C}{\left(u)}^{p}=\mathop{\int }\limits_{{S}^{n-1}}{\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{+}^{p}{\rm{d}}{\mu }_{p,C}\left(c).Based on the fact that hC⋅u(v)=hC(u⋅v){h}_{C\cdot u}\left(v)={h}_{C}\left(u\cdot v)and the sesquilinearity of the Hermitian inner product in Cn{{\mathbb{C}}}^{n}, we obtain hC⋅u(v)p=∫Sn−1(ℛ[c⋅(u⋅v)])+pdμp,C(c)=∫Sn−1(ℛ[cu⋅v])+pdμp,C(c){h}_{C\cdot u}{\left(v)}^{p}=\mathop{\int }\limits_{{S}^{n-1}}{\left({\mathcal{ {\mathcal R} }}\left[c\cdot \left(u\cdot v)])}_{+}^{p}{\rm{d}}{\mu }_{p,C}\left(c)=\mathop{\int }\limits_{{S}^{n-1}}{\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{+}^{p}{\rm{d}}{\mu }_{p,C}\left(c)\hspace{1.75em}for all u,v∈Sn−1u,v\in {S}^{n-1}. Then, if p≥1,K∈Ko(Cn)p\ge 1,K\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})is an asymmetric Lp{L}_{p}zonoid, the asymmetric complex Lp{L}_{p}projection body Πp,C+K{\Pi }_{p,C}^{+}Kis (1.3)h(Πp,C+K,u)p=2nVp(K,C⋅u)=∫S2n−1∫Sn−1(ℛ[cu⋅v])+pdμp,C(c)dSp,K(v)h{\left({\Pi }_{p,C}^{+}K,u)}^{p}=2n{V}_{p}\left(K,C\cdot u)=\mathop{\int }\limits_{{S}^{2n-1}}\mathop{\int }\limits_{{S}^{n-1}}{\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{+}^{p}{\rm{d}}{\mu }_{p,C}\left(c){\rm{d}}{S}_{p,K}\left(v)for all u,v∈S2n−1u,v\in {S}^{2n-1}, where Sp,K(v){S}_{p,K}\left(v)is the Lp{L}_{p}surface area measure of KKon S2n−1{S}^{2n-1}.Motivated by the works of Abardia and Bernig [22], Haberl [20], and Liu and Wang [24], we introduce more general definitions of complex Lp{L}_{p}projection body and complex Lp{L}_{p}mixed projection body.Definition 1.1If p≥1p\ge 1, K∈Ko(Cn)K\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})is an asymmetric Lp{L}_{p}zonoid, the general complex Lp{L}_{p}projection body Πp,CλK{\Pi }_{p,C}^{\lambda }Kis defined by (1.4)h(Πp,CλK,u)p=f1(λ)h(Πp,C+K,u)p+f2(λ)h(Πp,C−K,u)ph{\left({\Pi }_{p,C}^{\lambda }K,u)}^{p}={f}_{1}\left(\lambda )h{\left({\Pi }_{p,C}^{+}K,u)}^{p}+{f}_{2}\left(\lambda )h{\left({\Pi }_{p,C}^{-}K,u)}^{p}for every λ∈[−1,1]\lambda \in \left[-1,1], where f1(λ)=(1+λ)p(1+λ)p+(1−λ)p,f2(λ)=(1−λ)p(1+λ)p+(1−λ)p,{f}_{1}\left(\lambda )=\frac{{\left(1+\lambda )}^{p}}{{\left(1+\lambda )}^{p}+{\left(1-\lambda )}^{p}},\hspace{1em}{f}_{2}\left(\lambda )=\frac{{\left(1-\lambda )}^{p}}{{\left(1+\lambda )}^{p}+{\left(1-\lambda )}^{p}},and f1(λ)+f2(λ)=1{f}_{1}\left(\lambda )+{f}_{2}\left(\lambda )=1.We use Πp,Cλ,∗K{\Pi }_{p,C}^{\lambda ,\ast }Kto denote the polar body Πp,CλK{\Pi }_{p,C}^{\lambda }K. The normalization is chosen such that Πp,CλB=B{\Pi }_{p,C}^{\lambda }B=Band Πp,C0K=Πp,CK{\Pi }_{p,C}^{0}K={\Pi }_{p,C}K. If λ=1\lambda =1in (1.4), then Πp,C1K=Πp,C+K{\Pi }_{p,C}^{1}K={\Pi }_{p,C}^{+}K. In addition, set Πp,C−K=Πp,C+(−K){\Pi }_{p,C}^{-}K={\Pi }_{p,C}^{+}\left(-K). By the definitions of Πp,C±K{\Pi }_{p,C}^{\pm }Kand Πp,CλK{\Pi }_{p,C}^{\lambda }K, we obtain (1.5)Πp,CλK=(1+λ)p(1+λ)p+(1−λ)p⋅Πp,C+K+p(1−λ)p(1+λ)p+(1−λ)p⋅Πp,C−K,{\Pi }_{p,C}^{\lambda }K=\frac{{\left(1+\lambda )}^{p}}{{\left(1+\lambda )}^{p}+{\left(1-\lambda )}^{p}}\cdot {\Pi }_{p,C}^{+}K{+}_{p}\frac{{\left(1-\lambda )}^{p}}{{\left(1+\lambda )}^{p}+{\left(1-\lambda )}^{p}}\cdot {\Pi }_{p,C}^{-}K,and the complex Lp{L}_{p}projection body is defined as (1.6)Πp,CK=12⋅Πp,C+K+p12⋅Πp,C−K.{\Pi }_{p,C}K=\frac{1}{2}\cdot {\Pi }_{p,C}^{+}K{+}_{p}\frac{1}{2}\cdot {\Pi }_{p,C}^{-}K.It is clear that if p=1p=1in (1.6), Πp,CK{\Pi }_{p,C}Kis ΠCK{\Pi }_{C}Kdefined in (1.1).Definition 1.2If p≥1p\ge 1, K1,K2,…,K2n−1∈Ko(Cn){K}_{1},{K}_{2},\ldots ,{K}_{2n-1}\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})is an asymmetric Lp{L}_{p}zonoid, the general complex Lp{L}_{p}mixed projection body Πp,Cλ(K1,K2,…,K2n−1){\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})is defined by (1.7)h(Πp,Cλ(K1,K2,…,K2n−1),u)p=f1(λ)h(Πp,C+(K1,K2,…,K2n−1),u)p+f2(λ)h(Πp,C−(K1,K2,…,K2n−1),u)p\hspace{1.8em}h{\left({\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),u)}^{p}={f}_{1}\left(\lambda )h{\left({\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),u)}^{p}+{f}_{2}\left(\lambda )h{\left({\Pi }_{p,C}^{-}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),u)}^{p}for every λ∈[−1,1]\lambda \in \left[-1,1].Moreover, if K2n−i=⋯=K2n−1=B{K}_{2n-i}=\cdots ={K}_{2n-1}=Band M≔(K1,K2,…,K2n−1−i){\bf{M}}:= \left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1-i}), then for 0≤i≤2n−20\le i\le 2n\hspace{-0.02em}-\hspace{-0.02em}2, Πp,Cλ(K1,K2,…,K2n−1){\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})is written as Πp,i,Cλ(M){\Pi }_{p,i,C}^{\lambda }\left({\bf{M}}). If K1=⋯=K2n−1−i=K{K}_{1}=\cdots ={K}_{2n-1-i}=K, we simply write Πp,i,Cλ(M){\Pi }_{p,i,C}^{\lambda }\left({\bf{M}})as Πp,i,Cλ(K){\Pi }_{p,i,C}^{\lambda }\left(K)that is called the iith Lp{L}_{p}mixed complex projection body of KK. If i=0i=0, we write Πp,0,CλK{\Pi }_{p,0,C}^{\lambda }Kas Πp,CλK{\Pi }_{p,C}^{\lambda }K.Before stating our main results, let us introduce the Lp{L}_{p}Blaschke combination. For 2n≠p≥12n\ne p\ge 1and K,L∈Kos(Cn)K,L\in {{\mathcal{K}}}_{os}\left({{\mathbb{C}}}^{n}), the Lp{L}_{p}Blaschke combination K#pL∈Kos(Cn)K\hspace{-0.3em}{\text{\#}}_{p}L\in {{\mathcal{K}}}_{os}\left({{\mathbb{C}}}^{n})is defined by Lutwak [4] as (1.8)dSp(K#pL,⋅)≔dSp(K,⋅)+dSp(L,⋅),{\rm{d}}{S}_{p}\left(K\hspace{-0.3em}{\text{\#}}_{p}L,\cdot ):= {\rm{d}}{S}_{p}\left(K,\cdot )+{\rm{d}}{S}_{p}\left(L,\cdot ),where Sp(K,⋅){S}_{p}\left(K,\cdot )denotes the Lp{L}_{p}surface area measure of KKon S2n−1{S}^{2n-1}. If p=1p=1and K,L∈K(Cn)K,L\in {\mathcal{K}}\left({{\mathbb{C}}}^{n}), it is a classical Blaschke combination.Our main results can be stated as the following Theorems 1.1–1.4 and among them, Theorems 1.1–1.2 are the Brunn-Minkowski-type inequalities for the general complex Lp{L}_{p}projection bodies. Theorems 1.3–1.4 are the Aleksandrov-Fenchel-type inequalities for the general complex Lp{L}_{p}mixed projection bodies.Theorem 1.1If p≥1p\ge 1, K,L∈Kos(Cn)K,L\in {{\mathcal{K}}}_{os}\left({{\mathbb{C}}}^{n}), and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})is an asymmetric Lp{L}_{p}zonoid, then(1.9)V(Πp,Cλ(K#pL))p2n≥V(Πp,CλK)p2n+V(Πp,CλL)p2nV{\left({\Pi }_{p,C}^{\lambda }\left(K{\text{\#}}_{p}L))}^{\tfrac{p}{2n}}\ge V{\left({\Pi }_{p,C}^{\lambda }K)}^{\tfrac{p}{2n}}+V{\left({\Pi }_{p,C}^{\lambda }L)}^{\tfrac{p}{2n}}with equality if and only if KKand LLare real dilates.Theorem 1.2If p≥1p\ge 1, K,L∈Kos(Cn)K,L\in {{\mathcal{K}}}_{os}\left({{\mathbb{C}}}^{n}), and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})is an asymmetric Lp{L}_{p}zonoid, then(1.10)V(Πp,Cλ,∗(K#pL))−p2n≥V(Πp,Cλ,∗K)−p2n+V(Πp,Cλ,∗L)−p2nV{\left({\Pi }_{p,C}^{\lambda ,\ast }\left(K{\text{\#}}_{p}L))}^{-\tfrac{p}{2n}}\ge V{\left({\Pi }_{p,C}^{\lambda ,\ast }K)}^{-\tfrac{p}{2n}}+V{\left({\Pi }_{p,C}^{\lambda ,\ast }L)}^{-\tfrac{p}{2n}}with equality if and only if KKand LLare real dilates.Theorem 1.3If p≥1p\ge 1, K1,K2,…,K2n−1∈Ko(Cn){K}_{1},{K}_{2},\ldots ,{K}_{2n-1}\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})is an asymmetric Lp{L}_{p}zonoid, then(1.11)V(Πp,Cλ(K1,K2,…,K2n−1))pr2n≥(2n)1−p∏j=12n−1V(K1,…,K2n−1,Kj)r(1−p)2n−1∏j=1rV(ΠC(Kj[r],Kr+1,…,K2n−1))p2n.V{\left({\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{pr}{2n}}\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{r\left(1-p)}{2n-1}}\mathop{\prod }\limits_{j=1}^{r}V{\left({\Pi }_{C}\left({K}_{j}\left[r],{K}_{r+1},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n}}.If Kj{K}_{j}is an ellipsoid centered at the origin or an Hermitian ellipsoid, the equality holds.Theorem 1.4If K1,K2,…,K2n−1−i∈Ko(Cn){K}_{1},{K}_{2},\ldots ,{K}_{2n-1-i}\hspace{-0.03em}\in \hspace{-0.03em}{{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n})and C∈K(C)C\hspace{-0.03em}\in \hspace{-0.03em}{\mathcal{K}}\left({\mathbb{C}})is an asymmetric Lp{L}_{p}zonoid. Let M≔(K1,K2,…,K2n−1−i){\bf{M}}\hspace{-0.03em}:= \hspace{-0.03em}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1-i}). If p≥1p\ge 1, i=0,1,…,2n−2i=0,1,\ldots ,2n-2, then(1.12)V(Πp,i,CλM)p2n≥(2n)1−p∏j=12n−1−iWi(M,Kj)1−p2n−1−i∏j=1rV(Πi,C(Kj[r],Kr+1,…,K2n−1−i))p2nr.V{\left({\Pi }_{p,i,C}^{\lambda }{\bf{M}})}^{\tfrac{p}{2n}}\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1-i}{W}_{i}{\left({\bf{M}},{K}_{j})}^{\tfrac{1-p}{2n-1-i}}\mathop{\prod }\limits_{j=1}^{r}V{\left({\Pi }_{i,C}\left({K}_{j}\left[r],{K}_{r+1},\ldots ,{K}_{2n-1-i}))}^{\tfrac{p}{2nr}}.If Kj{K}_{j}is an ellipsoid centered at the origin or an Hermitian ellipsoid, the equality holds.Remark 1.1Note that the cases of C=[0,1]C=\left[0,1]of Theorems 1.1–1.4 are the Brunn-Minkowski-type inequalities [26] and the Aleksandrov-Fenchel-type inequalities for the general Lp{L}_{p}projection bodies [27].If CCis just a point, then Πp,CλK={0}{\Pi }_{p,C}^{\lambda }K=\{0\}for every K∈Ko(Cn)K\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n})and every λ∈[−1,1]\lambda \in \left[-1,1]. We assume that dimC>0\hspace{0.1em}\text{dim}\hspace{0.1em}\hspace{0.33em}C\gt 0throughout this paper.2Preliminaries2.1Support function, radial function, and polar of convex bodyFor a complex number c∈Cc\in {\mathbb{C}}, we write c¯\bar{c}for its complex conjugate and ∣c∣| c| for its norm. If ϕ∈Cm×n\phi \in {{\mathbb{C}}}^{m\times n}, then ϕ∗{\phi }^{\ast }denotes the conjugate transpose of ϕ\phi . We denote by “⋅\cdot ” the standard Hermitian inner product on Cn{{\mathbb{C}}}^{n}which is conjugate linear in first argument. Koldobsky et al. [19] identified Cn{{\mathbb{C}}}^{n}with R2n{{\mathbb{R}}}^{2n}using the standard mapping: ι(c)=(ℛ[c1],…,ℛ[cn],ζ[c1],…,ζ[cn]),c∈Cn,\iota \left(c)=\left({\mathcal{ {\mathcal R} }}\left[{c}_{1}],\ldots ,{\mathcal{ {\mathcal R} }}\left[{c}_{n}],\zeta \left[{c}_{1}],\ldots ,\zeta \left[{c}_{n}]),\hspace{1em}c\in {{\mathbb{C}}}^{n},\hspace{1.35em}where ℛ{\mathcal{ {\mathcal R} }}and ζ\zeta are the real and imaginary parts, respectively. Note that ℛ[x⋅y]=ιx⋅ιy{\mathcal{ {\mathcal R} }}\left[x\cdot y]=\iota x\cdot \iota yfor all x,y∈Cnx,y\in {{\mathbb{C}}}^{n}, where the inner product on the right hand is the standard Euclidean inner product on R2n{{\mathbb{R}}}^{2n}.We collect complex reformulations of well-known results from convex geometry. These complex version can be directly deduced from their real counterparts by an appropriate application of ι\iota . For more details refer to the books in [28,29].A convex body K∈K(Cn)K\in {\mathcal{K}}\left({{\mathbb{C}}}^{n})is determined by its support function hK:Cn→R{h}_{K}:{{\mathbb{C}}}^{n}\to {\mathbb{R}}, where hK(x)=max{ℛ[x⋅y]:y∈K}.{h}_{K}\left(x)=\hspace{0.1em}\text{max}\hspace{0.1em}\{{\mathcal{ {\mathcal R} }}\left[x\cdot y]:y\in K\}.For every Borel set ω⊂S2n−1\omega \subset {S}^{2n-1}, the complex surface area measure SK{S}_{K}of K∈K(Cn)K\in {\mathcal{K}}\left({{\mathbb{C}}}^{n})is defined by SK(ω)=ℋ2n−1(ι{x∈K:∃u∈ω,ℛ[x⋅u]=hK(u)}),{S}_{K}\left(\omega )={{\mathcal{ {\mathcal H} }}}^{2n-1}\left(\iota \{x\in K:\exists u\in \omega ,{\mathcal{ {\mathcal R} }}\left[x\cdot u]={h}_{K}\left(u)\}),where ℋ2n−1{{\mathcal{ {\mathcal H} }}}^{2n-1}stands for (2n−12n-1)-dimensional Hausdorff measure on R2n{{\mathbb{R}}}^{2n}. In addition, the complex surface area measures are translation invariant and ScK(ω)=SK(c¯ω){S}_{cK}\left(\omega )={S}_{K}\left(\bar{c}\omega )for all c∈Sn−1c\in {S}^{n-1}and each Borel set ω∈S2n−1\omega \in {S}^{2n-1}.KKis an Hermitian ellipsoid if K={x∈Cn:x⋅ϕx≤1}+tK=\left\{x\in {{\mathbb{C}}}^{n}:x\cdot \phi x\le 1\right\}+tfor a positive definite Hermitian matrix ϕ∈GL(n,C)\phi \in GL\left(n,{\mathbb{C}})and a t∈Cnt\in {{\mathbb{C}}}^{n}. Note that if KKis an Hermitian ellipsoid if and only if K=ψB+tK=\psi B+tfor some ψ∈GL(n,C)\psi \in GL\left(n,{\mathbb{C}})and a t∈Cnt\in {{\mathbb{C}}}^{n}(see [20]).If KKis a compact star-shaped (about the origin) in Cn{{\mathbb{C}}}^{n}, its radial function, ρK(x)=ρ(K,x):Cn⧹{0}→[0,∞){\rho }_{K}\left(x)=\rho \left(K,x):{{\mathbb{C}}}^{n}\setminus \{0\}\to {[}0,\infty )is given by ρK(x)=max{λ≥0:λx∈K}.{\rho }_{K}\left(x)=\hspace{0.1em}\text{max}\hspace{0.1em}\{\lambda \ge 0:\lambda x\in K\}.If ρK(x){\rho }_{K}\left(x)is positive and continuous, KKwill be called a star body. For the set of star body containing the origin in their interiors, we write So(Cn){{\mathcal{S}}}_{o}\left({{\mathbb{C}}}^{n}). Moreover, if K∈Ko(Cn)K\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), then K∗∈Ko(Cn){K}^{\ast }\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}). On Cn⧹{0}{{\mathbb{C}}}^{n}\setminus \{0\}, the support function and radial function of the polar body K∗{K}^{\ast }can be given, respectively, by hK∗=1ρK,ρK∗=1hK.{h}_{{K}^{\ast }}=\frac{1}{{\rho }_{K}},\hspace{1em}{\rho }_{{K}^{\ast }}=\frac{1}{{h}_{K}}.\hspace{3.2em}2.2The Lp{L}_{p}mixed quermassintegralsFor K∈Ko(Cn)K\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), i=1,2,…,2n−1i=1,2,\ldots ,2n-1, the quermassintegrals Wi(K){W}_{i}\left(K)of KKare defined by (see [29]) (2.1)Wi(K)=12n∫S2n−1h(K,u)dSi(K,u),{W}_{i}\left(K)=\frac{1}{2n}\mathop{\int }\limits_{{S}^{2n-1}}h\left(K,u){\rm{d}}{S}_{i}\left(K,u),where Si(K,u){S}_{i}\left(K,u)is the mixed surface area of KK.If p≥1,K,L∈Ko(Cn)p\ge 1,K,L\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), and α,β≥0\alpha ,\beta \ge 0that are not both zero, the Lp{L}_{p}Minkowski combination α⋅K+pβ⋅L\alpha \cdot K{+}_{p}\beta \cdot Lis defined by hα⋅K+pβ⋅Lp=αhKp+βhLp.{h}_{\alpha \cdot K{+}_{p}\beta \cdot L}^{p}=\alpha {h}_{K}^{p}+\beta {h}_{L}^{p}.The extension of the Brunn-Minkowski inequality (see [4]) is as follows: If K,L∈Ko(Cn),p>1K,L\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}),p\gt 1, then (2.2)Wi(K+pL)p2n−i≥Wi(K)p2n−i+Wi(L)p2n−i{W}_{i}{\left(K{+}_{p}L)}^{\tfrac{p}{2n-i}}\ge {W}_{i}{\left(K)}^{\tfrac{p}{2n-i}}+{W}_{i}{\left(L)}^{\tfrac{p}{2n-i}}with equality if and only if KKand LLare dilates. In particular, if p=1p=1, i=0i=0, the inequality is the Brunn-Minkowski inequality.If p≥1p\ge 1, i=1,2,…,2n−1i=1,2,\ldots ,2n-1, and K∈Ko(Cn)K\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), there exists a positive Borel measure Sp,i(K,⋅){S}_{p,i}\left(K,\cdot )on S2n−1{S}^{2n-1}, such that the Lp{L}_{p}mixed quermassintegral Wp,i(K,L){W}_{p,i}\left(K,L)has the following integral representation (see [4]): (2.3)Wp,i(K,L)=12n∫S2n−1h(L,u)pdSp,i(K,u){W}_{p,i}\left(K,L)=\frac{1}{2n}\mathop{\int }\limits_{{S}^{2n-1}}h{\left(L,u)}^{p}{\rm{d}}{S}_{p,i}\left(K,u)for all L∈Ko(Cn)L\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}). It turns out that the measure Sp,i(K,⋅){S}_{p,i}\left(K,\cdot )on S2n−1{S}^{2n-1}is absolutely continuous with respect to Si(K,⋅){S}_{i}\left(K,\cdot )and has the Radon-Nikodym derivative dSp,i(K,⋅)dSi(K,⋅)=hK(⋅)1−p\frac{{\rm{d}}{S}_{p,i}\left(K,\cdot )}{{\rm{d}}{S}_{i}\left(K,\cdot )}={h}_{K}{\left(\cdot )}^{1-p}. Obviously, Sp,0(K,⋅)=Sp(K,⋅){S}_{p,0}\left(K,\cdot )={S}_{p}\left(K,\cdot ).In view of the Lp{L}_{p}Minkowski inequality in Rn{{\mathbb{R}}}^{n}by Lutwak [4], there is an Lp{L}_{p}Minkowski inequality about the Lp{L}_{p}mixed quermassintegrals in Cn{{\mathbb{C}}}^{n}. That is, if K,L∈Ko(Cn),p≥1K,L\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}),p\ge 1, and 0≤i<2n0\le i\lt 2n, then (2.4)Wp,i(K,L)2n−i≥Wi(K)2n−i−pWi(L)p{W}_{p,i}{\left(K,L)}^{2n-i}\ge {W}_{i}{\left(K)}^{2n-i-p}{W}_{i}{\left(L)}^{p}with equality for p=1p=1if and only if KKand LLare real homothetic, while for p>1p\gt 1if and only if KKand LLare real dilates. In particular, if i=0i=0in (2.4), then (2.5)Vp(K,L)2n≥V(K)2n−pV(L)p{V}_{p}{\left(K,L)}^{2n}\ge V{\left(K)}^{2n-p}V{\left(L)}^{p}with equality if and only if KKand LLare real dilates.2.3The dual Lp{L}_{p}mixed quermassintegralsFor K∈So(Cn)K\in {{\mathcal{S}}}_{o}\left({{\mathbb{C}}}^{n})and i=1,2,…,2n−1i=1,2,\ldots ,2n-1, the dual quermassintegrals W˜i(K){\widetilde{W}}_{i}\left(K)are defined by (see [30]) (2.6)W˜i(K)=12n∫S2n−1ρ(K,u)2n−idS(u),{\widetilde{W}}_{i}\left(K)=\frac{1}{2n}\mathop{\int }\limits_{{S}^{2n-1}}\rho {\left(K,u)}^{2n-i}{\rm{d}}S\left(u),where S(u)S\left(u)stands for the push forward with respect to ι−1{\iota }^{-1}of ℋ2n−1{{\mathcal{ {\mathcal H} }}}^{2n-1}on the (2n−1)\left(2n-1)-dimensional Euclidean unit sphere.If p≥1,K,L∈So(Cn)p\ge 1,K,L\in {{\mathcal{S}}}_{o}\left({{\mathbb{C}}}^{n}), and α,β≥0\alpha ,\beta \ge 0that are not both zero, the Lp{L}_{p}harmonic radial combination α⋅K+˜pβ⋅L\alpha \cdot K{\widetilde{+}}_{p}\beta \cdot Lis defined by ρα⋅K+˜pβ⋅L−p=αρK−p+βρL−p.{\rho }_{\alpha \cdot K{\widetilde{+}}_{p}\beta \cdot L}^{-p}=\alpha {\rho }_{K}^{-p}+\beta {\rho }_{L}^{-p}.For p≥1,i=1,2,…,2n−1p\ge 1,i=1,2,\ldots ,2n-1, and K,L∈So(Cn)K,L\in {{\mathcal{S}}}_{o}\left({{\mathbb{C}}}^{n}), the dual Lp{L}_{p}mixed quermassintegral W˜p,i(K,L){\widetilde{W}}_{p,i}\left(K,L)has the following integral representation (see [31]): (2.7)W˜−p,i(K,L)=12n∫S2n−1ρ(K,u)2n+p−iρ(L,u)−pdS(u).{\widetilde{W}}_{-p,i}\left(K,L)=\frac{1}{2n}\mathop{\int }\limits_{{S}^{2n-1}}\rho {\left(K,u)}^{2n+p-i}\rho {\left(L,u)}^{-p}{\rm{d}}S\left(u).The dual Lp{L}_{p}Minkowski inequality in Cn{{\mathbb{C}}}^{n}can be stated as follows: If p≥1p\ge 1, K,L∈So(Cn)K,L\in {{\mathcal{S}}}_{o}\left({{\mathbb{C}}}^{n}), and 0≤i<2n0\le i\lt 2n, then (2.8)W˜−p,i(K,L)2n−i≥W˜i(K)2n+p−iW˜i(L)−p{\widetilde{W}}_{-p,i}{\left(K,L)}^{2n-i}\ge {\widetilde{W}}_{i}{\left(K)}^{2n+p-i}{\widetilde{W}}_{i}{\left(L)}^{-p}with equality if and only if KKand LLare real dilates.2.4The general complex Lp{L}_{p}projection body and complex Lp{L}_{p}mixed projection bodySince the integral representations of the general complex Lp{L}_{p}projection body and complex Lp{L}_{p}mixed projection body need to be used in Section 3, we will present their integral representations in this part.First of all, by combining (1.3) and (1.4), we obtain the integral representation of the general complex Lp{L}_{p}projection body Πp,CλK{\Pi }_{p,C}^{\lambda }Kas (2.9)h(Πp,CλK,u)p=∫S2n−1∫Sn−1f1(λ)(ℛ[cu⋅v])+pdμp,C(c)dSp(K,v)+∫S2n−1∫Sn−1f2(λ)(ℛ[cu⋅v])−pdμp,C(c)dSp(K,v)h{\left({\Pi }_{p,C}^{\lambda }K,u)}^{p}=\mathop{\int }\limits_{{S}^{2n-1}}\mathop{\int }\limits_{{S}^{n-1}}{f}_{1}\left(\lambda ){\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{+}^{p}{\rm{d}}{\mu }_{p,C}\left(c){\rm{d}}{S}_{p}\left(K,v)+\mathop{\int }\limits_{{S}^{2n-1}}\mathop{\int }\limits_{{S}^{n-1}}{f}_{2}\left(\lambda ){\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{-}^{p}{\rm{d}}{\mu }_{p,C}\left(c){\rm{d}}{S}_{p}\left(K,v)for all u∈S2n−1u\in {S}^{2n-1}and every λ∈[−1,1]\lambda \in \left[-1,1].In order to give the integral representation of Πp,Cλ(K1,K2,…,K2n−1){\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}), we introduce the following definition.Definition 2.1For p≥0p\ge 0and K1,K2,…,K2n−1∈Ko(Cn){K}_{1},{K}_{2},\ldots ,{K}_{2n-1}\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), we define Borel measure Sp(K1,K2,…,K2n−1,⋅){S}_{p}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1},\cdot )on S2n−1{S}^{2n-1}as (2.10)Sp(K1,K2,…,K2n−1;ω)=∫ω(h(K1,u)h(K2,u)…h(K2n−1,u))1−p2n−1dS(K1,K2,…,K2n−1;u){S}_{p}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}\omega )=\mathop{\int }\limits_{\omega }{\left(h\left({K}_{1},u)h\left({K}_{2},u)\ldots h\left({K}_{2n-1},u))}^{\tfrac{1-p}{2n-1}}{\rm{d}}S\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}u)for each Borel ω⊂S2n−1\omega \subset {S}^{2n-1}.Let us mention that Sp(K1,K2,…,K2n−1;⋅){S}_{p}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}\cdot )on S2n−1{S}^{2n-1}is called the general complex Lp{L}_{p}mixed surface area measure of K1,K2,…,K2n−1{K}_{1},{K}_{2},\ldots ,{K}_{2n-1}and has the Radon-Nikodym derivative (2.11)dSp(K1,K2,…,K2n−1;⋅)dS(K1,K2,…,K2n−1;⋅)=(h(K1,⋅)h(K2,⋅)…h(K2n−1,⋅))1−p2n−1.\hspace{-26.85em}\frac{{\rm{d}}{S}_{p}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}\cdot )}{{\rm{d}}S\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}\cdot )}={\left(h\left({K}_{1},\cdot )h\left({K}_{2},\cdot )\ldots h\left({K}_{2n-1},\cdot ))}^{\tfrac{1-p}{2n-1}}.If K1=K2=⋯=K2n−1=K{K}_{1}={K}_{2}=\cdots ={K}_{2n-1}=K, then Sp(K,K,…,K,⋅)=Sp(K,⋅){S}_{p}\left(K,K,\ldots ,K,\cdot )={S}_{p}\left(K,\cdot ). In particular, if K1,K2,…,K2n−1−i∈Ko(Cn),K2n−i=⋯=K2n−1=B{K}_{1},{K}_{2},\ldots ,{K}_{2n-1-i}\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}),{K}_{2n-i}=\cdots ={K}_{2n-1}=B, then for i=0,1,…,2n−2i=0,1,\ldots ,2n-2, we obtain (2.12)dSp,i(K1,K2,…,K2n−1−i;⋅)dSi(K1,K2,…,K2n−1−i;⋅)=(h(K1,⋅)h(K2,⋅)…h(K2n−1−i,⋅))1−p2n−1−i.\frac{{\rm{d}}{S}_{p,i}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1-i};\hspace{0.33em}\cdot )}{{\rm{d}}{S}_{i}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1-i};\hspace{0.33em}\cdot )}={\left(h\left({K}_{1},\cdot )h\left({K}_{2},\cdot )\ldots h\left({K}_{2n-1-i},\cdot ))}^{\tfrac{1-p}{2n-1-i}}.Next, with respect to (1.7) and (2.10), we have the following integral representation of the general complex Lp{L}_{p}mixed projection body Πp,Cλ(K1,K2,…,K2n−1){\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})as (2.13)h(Πp,Cλ(K1,K2,…,K2n−1),u)p=∫S2n−1∫Sn−1f1(λ)(ℛ[cu⋅v])+pdSp(K1,K2,…,K2n−1;v)+∫S2n−1∫Sn−1f2(λ)(ℛ[cu⋅v])+pdSp(K1,K2,…,K2n−1;v)\begin{array}{rcl}h{\left({\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),u)}^{p}& =& \mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{n-1}}{f}_{1}\left(\lambda ){\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{+}^{p}{\rm{d}}{S}_{p}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}v)\\ & & +\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{n-1}}{f}_{2}\left(\lambda ){\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{+}^{p}{\rm{d}}{S}_{p}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}v)\end{array}for all u,v∈S2n−1u,v\in {S}^{2n-1}and every λ∈[−1,1]\lambda \in \left[-1,1].3Proofs of main resultsIn this section, we give the proofs of Theorems 1.1–1.4. First, the proof of Theorem 1.1 needs the following Lemma 3.1.Lemma 3.1Let K∈Ko(Cn)K\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n})and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})be an asymmetric Lp{L}_{p}zonoid. If p≥1p\ge 1and 0≤i<2n0\le i\lt 2n, then for all Q∈Ko(Cn)Q\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), we have(3.1)Wp,i(Q,Πp,CλK)=Vp(K,Πp,i,C¯λQ).{W}_{p,i}\left(Q,{\Pi }_{p,C}^{\lambda }K)={V}_{p}\left(K,{\Pi }_{p,i,\bar{C}}^{\lambda }Q).ProofFrom (2.3), (1.4), and the conjugate linear of Hermitian inner product, we know that for all Q∈Ko(Cn)Q\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), Wp,i(Q,Πp,CλK)=12n∫S2n−1h(Πp,CλK,u)pdSp,i(Q,u)=12n∫S2n−1∫S2n−1∫Sn−1f1(λ)(ℛ[cu⋅v])+pdμp,C(c)dSp(K,v)dSp,i(Q,u)+12n∫S2n−1∫S2n−1∫Sn−1f2(λ)(ℛ[cu⋅v])−pdμp,C(c)dSp(K,v)dSp,i(Q,u)=12n∫S2n−1∫S2n−1∫Sn−1f1(λ)(ℛ[u⋅c¯v])+pdμp,C(c)dSp,i(Q,u)Sp(K,v)+12n∫S2n−1∫S2n−1∫Sn−1f2(λ)(ℛ[u⋅c¯v])−pdμp,C(c)dSp,i(Q,u)Sp(K,v)=12n∫S2n−1∫S2n−1f1(λ)h(C¯u,v)pdSp,i(Q,u)dSp(K,v)+12n∫S2n−1∫S2n−1f2(λ)h(−C¯u,v)pdSp,i(Q,u)dSp(K,v)=12n∫S2n−1h(Πp,i,C¯λQ,v)pdSp(K,v)=Vp(K,Πp,i,C¯λQ),\begin{array}{rcl}{W}_{p,i}\left(Q,{\Pi }_{p,C}^{\lambda }K)& =& \frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}h{\left({\Pi }_{p,C}^{\lambda }K,u)}^{p}{\rm{d}}{S}_{p,i}\left(Q,u)\\ & =& \frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{n-1}}{f}_{1}\left(\lambda ){\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{+}^{p}{\rm{d}}{\mu }_{p,C}\left(c){\rm{d}}{S}_{p}\left(K,v){\rm{d}}{S}_{p,i}\left(Q,u)\\ & & +\frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{n-1}}{f}_{2}\left(\lambda ){\left({\mathcal{ {\mathcal R} }}\left[cu\cdot v])}_{-}^{p}{\rm{d}}{\mu }_{p,C}\left(c){\rm{d}}{S}_{p}\left(K,v){\rm{d}}{S}_{p,i}\left(Q,u)\\ & =& \frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{n-1}}{f}_{1}\left(\lambda ){\left({\mathcal{ {\mathcal R} }}\left[u\cdot \bar{c}v])}_{+}^{p}{\rm{d}}{\mu }_{p,C}\left(c){\rm{d}}{S}_{p,i}\left(Q,u){S}_{p}\left(K,v)\\ & & +\frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{n-1}}{f}_{2}\left(\lambda ){\left({\mathcal{ {\mathcal R} }}\left[u\cdot \bar{c}v])}_{-}^{p}{\rm{d}}{\mu }_{p,C}\left(c){\rm{d}}{S}_{p,i}\left(Q,u){S}_{p}\left(K,v)\\ & =& \frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}{f}_{1}\left(\lambda )h{\left(\bar{C}u,v)}^{p}{\rm{d}}{S}_{p,i}\left(Q,u){\rm{d}}{S}_{p}\left(K,v)\\ & & +\frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}{f}_{2}\left(\lambda )h{\left(-\bar{C}u,v)}^{p}{\rm{d}}{S}_{p,i}\left(Q,u){\rm{d}}{S}_{p}\left(K,v)\\ & =& \frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}h{\left({\Pi }_{p,i,\bar{C}}^{\lambda }Q,v)}^{p}{\rm{d}}{S}_{p}\left(K,v)\\ & =& {V}_{p}\left(K,{\Pi }_{p,i,\bar{C}}^{\lambda }Q),\end{array}where C¯\bar{C}is the conjugate of CCand then we conclude the proof.□Now, we are in a position to prove Theorem 1.1.Proof of Theorem 1.1Since K,L∈Kos(Cn)K,L\in {{\mathcal{K}}}_{os}\left({{\mathbb{C}}}^{n}), then for N∈Kos(Cn)N\in {{\mathcal{K}}}_{os}\left({{\mathbb{C}}}^{n})and by (1.8), we have (3.2)Vp(K#pL,N)=12n∫S2n−1h(N,u)pdSp(K#pL)=Vp(K,N)+Vp(L,N).{V}_{p}\left(K\hspace{-0.3em}{\text{\#}}_{p}L,N)=\frac{1}{2n}\mathop{\int }\limits_{{S}^{2n-1}}h{\left(N,u)}^{p}{\rm{d}}{S}_{p}\left(K\hspace{-0.3em}{\text{\#}}_{p}L)={V}_{p}\left(K,N)+{V}_{p}\left(L,N).According to Lemma 3.1, for all Q∈Kos(Cn)Q\in {{\mathcal{K}}}_{os}\left({{\mathbb{C}}}^{n}), one deduces Wp,i(Q,Πp,Cλ(K#pL))=Vp(K#pL,Πp,C¯λQ)=Vp(K,Πp,i,C¯λQ)+Vp(L,Πp,i,C¯λQ)=Wp,i(Q,Πp,CλK)+Wp,i(Q,Πp,CλL).\hspace{-27.5em}\begin{array}{rcl}{W}_{p,i}\left(Q,{\Pi }_{p,C}^{\lambda }\left(K\hspace{-0.3em}{\text{\#}}_{p}L))& =& {V}_{p}\left(K\hspace{-0.3em}{\text{\#}}_{p}L,{\Pi }_{p,\bar{C}}^{\lambda }Q)\\ & =& {V}_{p}\left(K,{\Pi }_{p,i,\bar{C}}^{\lambda }Q)+{V}_{p}\left(L,{\Pi }_{p,i,\bar{C}}^{\lambda }Q)\\ & =& {W}_{p,i}\left(Q,{\Pi }_{p,C}^{\lambda }K)+{W}_{p,i}\left(Q,{\Pi }_{p,C}^{\lambda }L).\end{array}By (2.4), we have (3.3)Wp,i(Q,Πp,Cλ(K#pL))≥Wi(Q)2n−p−i2n−iWi(Πp,CλK)p2n−i+Wi(Πp,CλL)p2n−i\hspace{-27.35em}{W}_{p,i}\left(Q,{\Pi }_{p,C}^{\lambda }\left(K\hspace{-0.3em}{\text{\#}}_{p}L))\ge {W}_{i}{\left(Q)}^{\tfrac{2n-p-i}{2n-i}}\left[{W}_{i}{\left({\Pi }_{p,C}^{\lambda }K)}^{\tfrac{p}{2n-i}}+{W}_{i}{\left({\Pi }_{p,C}^{\lambda }L)}^{\tfrac{p}{2n-i}}\right]with equality if and only if Q,KQ,K, and LLare real dilates.Taking Q=Πp,Cλ(K#pL)Q={\Pi }_{p,C}^{\lambda }\left(K\hspace{-0.3em}{\text{\#}}_{p}L)in (3.3), we obtain (3.4)Wi(Πp,Cλ(K#pL))p2n−i≥Wi(Πp,CλK)p2n−i+Wi(Πp,CλL)p2n−i\hspace{-27.35em}{W}_{i}{\left({\Pi }_{p,C}^{\lambda }\left(K{\text{\#}}_{p}L))}^{\tfrac{p}{2n-i}}\ge {W}_{i}{\left({\Pi }_{p,C}^{\lambda }K)}^{\tfrac{p}{2n-i}}+{W}_{i}{\left({\Pi }_{p,C}^{\lambda }L)}^{\tfrac{p}{2n-i}}with equality if and only KKand LLare real dilates.□The following lemma provides a connection of Πp,Cλ,∗{\Pi }_{p,C}^{\lambda ,\ast }and Mp,i,C¯λ{M}_{p,i,\bar{C}}^{\lambda }in terms of the dual Lp{L}_{p}mixed quermassintergral and their mixed volume. We need the lemma to prove Theorem 1.2.Lemma 3.2Let K∈Ko(Cn)K\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n})and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})be an asymmetric Lp{L}_{p}zonoid. If p≥1p\ge 1and 0≤i<2n0\le i\lt 2n, then for all Q∈So(Cn)Q\in {{\mathcal{S}}}_{o}\left({{\mathbb{C}}}^{n}), we have(3.5)W˜−p,i(Q,Πp,Cλ,∗K)=2n+p2Vp(K,Mp,i,C¯λQ),{\widetilde{W}}_{-p,i}\left(Q,{\Pi }_{p,C}^{\lambda ,\ast }K)=\frac{2n+p}{2}{V}_{p}\left(K,{M}_{p,i,\bar{C}}^{\lambda }Q),where Mp,CλQ{M}_{p,C}^{\lambda }Qis the general complex Lp{L}_{p}moment body [24]. If K2n−i=⋯=K2n−1=B{K}_{2n-i}=\cdots ={K}_{2n-1}=B, we write Mp,i,Cλ(Q,B){M}_{p,i,C}^{\lambda }\left(Q,B)as Mp,i,CλQ{M}_{p,i,C}^{\lambda }Q.ProofBy (2.7), (2.9), and the conjugate linear of Hermitian inner product, we have Vp(K,Mp,i,C¯λQ)=12n∫S2n−1h(Mp,i,C¯λQ,u)pdSp(K,u)=22n(2n+p)∫S2n−1∫S2n−1f1(λ)h(C¯u,v)pρ(Q,v)2n+p−idS(v)dSp(K,u)+22n(2n+p)∫S2n−1∫S2n−1f2(λ)h(−C¯u,v)pρ(Q,v)2n+p−idS(v)dSp(K,u)=22n(2n+p)∫S2n−1∫S2n−1f1(λ)h(Cu,v)pρ(Q,v)2n+p−idSp(K,u)dS(v)+22n(2n+p)∫S2n−1∫S2n−1f2(λ)h(−Cu,v)pρ(Q,v)2n+p−idS(v)dSp(K,u)=22n(2n+p)∫S2n−1h(Πp,CλK,v)pρ(Q,v)2n+p−idS(v)=22n(2n+p)∫S2n−1ρ(Πp,Cλ,∗K,v)−pρ(Q,v)2n+p−idS(v)=22n+pW˜−p,i(Q,Πp,Cλ,∗K),\begin{array}{rcl}{V}_{p}\left(K,{M}_{p,i,\bar{C}}^{\lambda }Q)& =& \frac{1}{2n}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}h{\left({M}_{p,i,\bar{C}}^{\lambda }Q,u)}^{p}{\rm{d}}{S}_{p}\left(K,u)\\ & =& \frac{2}{2n\left(2n+p)}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}{f}_{1}\left(\lambda )h{\left(\bar{C}u,v)}^{p}\rho {\left(Q,v)}^{2n+p-i}{\rm{d}}S\left(v){\rm{d}}{S}_{p}\left(K,u)\\ & & +\frac{2}{2n\left(2n+p)}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}{f}_{2}\left(\lambda )h{\left(-\bar{C}u,v)}^{p}\rho {\left(Q,v)}^{2n+p-i}{\rm{d}}S\left(v){\rm{d}}{S}_{p}\left(K,u)\\ & =& \frac{2}{2n\left(2n+p)}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}{f}_{1}\left(\lambda )h{\left(Cu,v)}^{p}\rho {\left(Q,v)}^{2n+p-i}{\rm{d}}{S}_{p}\left(K,u){\rm{d}}S\left(v)\\ & & +\frac{2}{2n\left(2n+p)}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}{f}_{2}\left(\lambda )h{\left(-Cu,v)}^{p}\rho {\left(Q,v)}^{2n+p-i}{\rm{d}}S\left(v){\rm{d}}{S}_{p}\left(K,u)\\ & =& \frac{2}{2n\left(2n+p)}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}h{\left({\Pi }_{p,C}^{\lambda }K,v)}^{p}\rho {\left(Q,v)}^{2n+p-i}{\rm{d}}S\left(v)\\ & =& \frac{2}{2n\left(2n+p)}\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}\rho {\left({\Pi }_{p,C}^{\lambda ,\ast }K,v)}^{-p}\rho {\left(Q,v)}^{2n+p-i}{\rm{d}}S\left(v)\\ & =& \frac{2}{2n+p}{\widetilde{W}}_{-p,i}\left(Q,{\Pi }_{p,C}^{\lambda ,\ast }K),\end{array}which ends the proof of Lemma 3.2.□Proof of Theorem 1.2From (2.8), (3.2), and Lemma 3.2, we have for all Q∈So(Cn)Q\in {{\mathcal{S}}}_{o}\left({{\mathbb{C}}}^{n})(3.6)W˜−p,i(Q,Πp,Cλ,∗(K#pL))=2n+p2Vp(K#pL,Mp,i,C¯λQ)=2n+p2[Vp(K,Mp,i,C¯λQ)+Vp(L,Mp,i,C¯λQ)]=W˜−p,i(Q,Πp,Cλ,∗K)+W˜−p,i(Q,Πp,Cλ,∗L)≥W˜i(Q)2n+p−i2n−iW˜i(Πp,Cλ,∗K)−p2n−i+W˜i(Πp,Cλ,∗L)−p2n−i\hspace{-34.75em}\begin{array}{rcl}{\widetilde{W}}_{-p,i}\left(Q,{\Pi }_{p,C}^{\lambda ,\ast }\left(K\hspace{-0.3em}{\text{\#}}_{p}L))& =& \frac{2n+p}{2}{V}_{p}\left(K\hspace{-0.3em}{\text{\#}}_{p}L,{M}_{p,i,\bar{C}}^{\lambda }Q)\\ & =& \frac{2n+p}{2}{[}{V}_{p}\left(K,{M}_{p,i,\bar{C}}^{\lambda }Q)+{V}_{p}\left(L,{M}_{p,i,\bar{C}}^{\lambda }Q)]\\ & =& {\widetilde{W}}_{-p,i}\left(Q,{\Pi }_{p,C}^{\lambda ,\ast }K)+{\widetilde{W}}_{-p,i}\left(Q,{\Pi }_{p,C}^{\lambda ,\ast }L)\\ & \ge & {\widetilde{W}}_{i}{\left(Q)}^{\tfrac{2n+p-i}{2n-i}}\left[{\widetilde{W}}_{i}{\left({\Pi }_{p,C}^{\lambda ,\ast }K)}^{-\tfrac{p}{2n-i}}+{\widetilde{W}}_{i}{\left({\Pi }_{p,C}^{\lambda ,\ast }L)}^{-\tfrac{p}{2n-i}}\right]\end{array}with equality if and only if Q,KQ,K, and LLare real dilates.Taking Q=Πp,Cλ,∗(K#pL)Q={\Pi }_{p,C}^{\lambda ,\ast }\left(K\hspace{-0.3em}{\text{\#}}_{p}L)in (3.6), it yields that (3.7)W˜i(Πp,Cλ,∗(K#pL))−p2n−i≥W˜i(Πp,Cλ,∗K)−p2n−i+W˜i(Πp,Cλ,∗L)−p2n−i{\widetilde{W}}_{i}{\left({\Pi }_{p,C}^{\lambda ,\ast }\left(K{\text{\#}}_{p}L))}^{-\tfrac{p}{2n-i}}\ge {\widetilde{W}}_{i}{\left({\Pi }_{p,C}^{\lambda ,\ast }K)}^{-\tfrac{p}{2n-i}}+{\widetilde{W}}_{i}{\left({\Pi }_{p,C}^{\lambda ,\ast }L)}^{-\tfrac{p}{2n-i}}with equality if and only if KKand LLare real dilates. Finally, let i=0i=0in (3.7), we conclude the proof of Theorem 1.2.□From now on, we pay attention to prove Theorems 1.3 and 1.4.Proof of Theorem 1.3From (1.3), (2.11), and the Hölder’s integral inequality [32], one has (3.8)h(Πp,C+(K1,K2,…,K2n−1),u)p=∫S2n−1h(C⋅u,v)pdSp(K1,K2,…,K2n−1;v)=∫S2n−1h(C⋅u,v)p(h(K1,u)⋯h(K2n−1,u))1−p2n−1dS(K1,K2,…,K2n−1;v)≥∫S2n−1h(C⋅u,v)dS(K1,K2,…,K2n−1;v)p(2n)1−p∏j=12n−1V(K1,K2,…,K2n−1,Kj)1−p2n−1=h(ΠC(K1,K2,…,K2n−1),u)p(2n)1−p∏j=12n−1V(K1,K2,…,K2n−1,Kj)1−p2n−1.\begin{array}{rcl}h{\left({\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),u)}^{p}& =& \mathop{\displaystyle \int }\limits_{{S}^{2n-1}}h{\left(C\cdot u,v)}^{p}{\rm{d}}{S}_{p}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}v)\\ & =& \mathop{\displaystyle \int }\limits_{{S}^{2n-1}}h{\left(C\cdot u,v)}^{p}{\left(h\left({K}_{1},u)\cdots h\left({K}_{2n-1},u))}^{\tfrac{1-p}{2n-1}}{\rm{d}}S\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};\hspace{0.33em}v)\\ & \ge & {\left(\mathop{\displaystyle \int }\limits_{{S}^{2n-1}}h\left(C\cdot u,v){\rm{d}}S\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1};v)\right)}^{p}{\left(2n)}^{1-p}\mathop{\displaystyle \prod }\limits_{j=1}^{2n-1}\\ & & V{\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}\\ & =& h{\left({\Pi }_{C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),u)}^{p}{\left(2n)}^{1-p}\mathop{\displaystyle \prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}.\end{array}According to the equality condition of Hölder’s integral inequality, the equality holds if and only if Kj{K}_{j}and C⋅uC\cdot uare dilates.For each Q∈Ko(Cn)Q\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), integrating both sides of (3.8) for dSp,i(Q,u){\rm{d}}{S}_{p,i}\left(Q,u)in u∈S2n−1u\in {S}^{2n-1}, we obtain Wp,i(Q,Πp,C+(K1,K2,…,K2n−1))≥(2n)1−p∏j=12n−1V(K1,K2,…,K2n−1,Kj)1−p2n−1Wp,i(Q,ΠC(K1,K2,…,K2n−1)).{W}_{p,i}\left(Q,{\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}{W}_{p,i}\left(Q,{\Pi }_{C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})).Taking Q=Πp,C+(K1,K2,…,K2n−1)Q={\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})and by (2.4), we have (3.9)Wi(Πp,C+(K1,K2,…,K2n−1))p2n−i≥(2n)1−p∏j=12n−1V(K1,K2,…,K2n−1,Kj)1−p2n−1Wi(ΠC(K1,K2,…,K2n−1))p2n−i\hspace{-39.85em}{W}_{i}{\left({\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n-i}}\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}{W}_{i}{\left({\Pi }_{C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n-i}}with equality if and only if Πp,C+(K1,K2,…,K2n−1),ΠC(K1,K2,…,K2n−1){\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),{\Pi }_{C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})are real dilates. But the equality condition in inequality (3.8) reveals that this happens precisely if Kj{K}_{j}and C⋅uC\cdot uare dilates.By the extension of the Brunn-Minkowski inequality (2.2), we obtain (3.10)Wi(Πp,Cλ(K1,K2,…,K2n−1))p2n−i≥Wi(Πp,C±(K1,K2,…,K2n−1))p2n−i{W}_{i}{\left({\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n-i}}\ge {W}_{i}{\left({\Pi }_{p,C}^{\pm }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n-i}}with equality if and only if Πp,C+(K1,K2,…,K2n−1),Πp,C−(K1,K2,…,K2n−1){\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}),{\Pi }_{p,C}^{-}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})are real dilates which is only possible if Πp,C−(K1,K2,…,K2n−1)=Πp,C+(K1,K2,…,K2n−1){\Pi }_{p,C}^{-}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})={\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}). It means that if Πp,C−(K1,K2,…,K2n−1)≠Πp,C+(K1,K2,…,K2n−1){\Pi }_{p,C}^{-}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1})\ne {\Pi }_{p,C}^{+}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}), the inequality is strict unless λ=−1,1\lambda =-1,1, or 0.Combining (3.9) and (3.10), we obtain (3.11)Wi(Πp,Cλ(K1,K2,…,K2n−1))p2n−i≥(2n)1−p∏j=12n−1V(K1,K2,…,K2n−1,Kj)1−p2n−1Wi(ΠC(K1,K2,…,K2n−1))p2n−i.{W}_{i}{\left({\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n-i}}\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}{W}_{i}{\left({\Pi }_{C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n-i}}.If p=1p=1, the Aleksandrov-Fenchel-type inequality (see [22]) is V(ΠC(K1,K2,…,K2n−1))r≥∏j=1rV(ΠC(Kj[r],Kr+1,…,K2n−1)).V{\left({\Pi }_{C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{r}\ge \mathop{\prod }\limits_{j=1}^{r}V\left({\Pi }_{C}\left({K}_{j}\left[r],{K}_{r+1},\ldots ,{K}_{2n-1})).After that, let i=0i=0in (3.11) and combine with the case of p=1p=1, we obtain (3.12)V(Πp,Cλ(K1,K2,…,K2n−1))p2n≥(2n)1−p∏j=12n−1V(K1,…,K2n−1,Kj)1−p2n−1∏j=1rV(ΠC(Kj[r],Kr+1,…,K2n−1))p2nr.V{\left({\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n}}\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}\mathop{\prod }\limits_{j=1}^{r}V{\left({\Pi }_{C}\left({K}_{j}\left[r],{K}_{r+1},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2nr}}.Let us turn toward the equality condition. If Kj{K}_{j}is an ellipsoid centered at the origin, then Kj=ϕB{K}_{j}=\phi Bfor ϕ∈GL(n,C)\phi \in GL\left(n,{\mathbb{C}}). From Πp,Cλ(ϕB)=∣detϕ∣2pϕ−∗Πp,CλB{\Pi }_{p,C}^{\lambda }\left(\phi B)=| {\rm{\det }}\hspace{0.33em}\phi \hspace{-0.25em}{| }^{\tfrac{2}{p}}{\phi }^{-\ast }{\Pi }_{p,C}^{\lambda }Band the fact that Πp,CλB=aB,ΠCB=bBa,b>0{\Pi }_{p,C}^{\lambda }B=aB,{\Pi }_{C}B=bB\hspace{0.33em}a,b\gt 0(see [20,24]), we have (3.13)V(Πp,Cλ(K1,K2,…,K2n−1))p2n=V(Πp,Cλ(ϕB))p2n=V(∣detϕ∣2pϕ−∗Πp,CλB)p2n=∣detϕ∣2V(ϕ−∗Πp,CλB)p2n=∣detϕ∣2∣detϕ−∗∣pnV(aB)p2n=∣detϕ∣2∣detϕ−∗∣pnapV(B)p2n.\hspace{-29.9em}\begin{array}{rcl}V{\left({\Pi }_{p,C}^{\lambda }\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n}}& =& V{\left({\Pi }_{p,C}^{\lambda }\left(\phi B))}^{\tfrac{p}{2n}}\\ & =& V{\left(| {\rm{\det }}\phi {| }^{\frac{2}{p}}{\phi }^{-\ast }{\Pi }_{p,C}^{\lambda }B)}^{\tfrac{p}{2n}}\\ & =& | {\rm{\det }}\hspace{0.33em}\phi \hspace{-0.25em}{| }^{2}V{\left({\phi }^{-\ast }{\Pi }_{p,C}^{\lambda }B)}^{\tfrac{p}{2n}}\\ & =& | {\rm{\det }}\hspace{0.33em}\phi \hspace{-0.25em}{| }^{2}| {\rm{\det }}\hspace{0.33em}{\phi }^{-\ast }\hspace{-0.25em}{| }^{\tfrac{p}{n}}V{\left(aB)}^{\tfrac{p}{2n}}\\ & =& | {\rm{\det }}\hspace{0.33em}\phi \hspace{-0.25em}{| }^{2}| {\rm{\det }}\hspace{0.33em}{\phi }^{-\ast }\hspace{-0.25em}{| }^{\tfrac{p}{n}}{a}^{p}V{\left(B)}^{\tfrac{p}{2n}}.\end{array}Similarly, we also obtain (3.14)V(ΠC(K1,K2,…,K2n−1))p2n=V(ΠC(ϕB))p2n=∣detϕ∣2p∣detϕ−∗∣pnbpV(B)p2n,V{\left({\Pi }_{C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n}}=V{\left({\Pi }_{C}\left(\phi B))}^{\tfrac{p}{2n}}=| {\rm{\det }}\hspace{0.33em}\phi \hspace{-0.25em}{| }^{2p}| {\rm{\det }}\hspace{0.33em}{\phi }^{-\ast }\hspace{-0.25em}{| }^{\tfrac{p}{n}}{b}^{p}V{\left(B)}^{\tfrac{p}{2n}},(3.15)(2n)1−p∏j=12n−1V(K1,…,K2n−1,Kj)1−p2n−1=(2n)1−pV(ϕB)1−p=(2n)1−p∣detϕ∣2(1−p)V(B)1−p=c∣detϕ∣2(1−p),{\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}={\left(2n)}^{1-p}V{\left(\phi B)}^{1-p}={\left(2n)}^{1-p}| {\rm{\det }}\hspace{0.33em}\phi \hspace{-0.25em}{| }^{2\left(1-p)}V{\left(B)}^{1-p}=c| {\rm{\det }}\hspace{0.33em}\phi \hspace{-0.25em}{| }^{2\left(1-p)},where c=∫S2n−1h(B,u)dS(B,u)1−pc={\left({\int }_{{S}^{2n-1}}h\left(B,u){\rm{d}}S\left(B,u)\right)}^{1-p}is a constant.From (3.14) and (3.15), we have (3.16)(2n)1−p∏j=12n−1V(K1,…,K2n−1,Kj)1−p2n−1∏j=1rV(ΠC(Kj[r],Kr+1,…,K2n−1))p2nr=c∣detϕ∣2∣detϕ−∗∣pnbpV(B)p2n.{\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}\mathop{\prod }\limits_{j=1}^{r}V{\left({\Pi }_{C}\left({K}_{j}\left[r],{K}_{r+1},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2nr}}=c| {\rm{\det }}\hspace{0.33em}\phi \hspace{-0.25em}{| }^{2}| {\rm{\det }}\hspace{0.33em}{\phi }^{-\ast }\hspace{-0.25em}{| }^{\tfrac{p}{n}}{b}^{p}V{\left(B)}^{\tfrac{p}{2n}}.Comparing equations (3.13) and (3.16), we know that a=c1pba={c}^{\tfrac{1}{p}}bis possible, which means that if Kj{K}_{j}is an ellipsoid centered at the origin, the equality holds in (3.12).If Kj{K}_{j}is an Hermitian ellipsoid, there exists a positive Hermitian matrix ϕ∈GL(n,C)\phi \in GL\left(n,{\mathbb{C}})and a vector t∈Cnt\in {{\mathbb{C}}}^{n}such that Kj=ϕB+t{K}_{j}=\phi B+t. The definition of Πp,Cλ{\Pi }_{p,C}^{\lambda }reveals that Πp,Cλ{\Pi }_{p,C}^{\lambda }is translation invariant. Hence, if Kj{K}_{j}is an Hermitian ellipsoid, equality also holds in (3.12).□The case of λ=0\lambda =0of Theorem 1.3 is the following Aleksandrov-Fenchel-type inequality for the complex Lp{L}_{p}mixed projection bodies.Corollary 3.1If p>1p\gt 1, K1,K2,…,K2n−1∈Ko(Cn){K}_{1},{K}_{2},\ldots ,{K}_{2n-1}\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), and C∈K(C)C\in {\mathcal{K}}\left({\mathbb{C}})is an asymmetric Lp{L}_{p}zonoid, thenV(Πp,C(K1,K2,…,K2n−1))p2n≥(2n)1−p∏j=12n−1V(K1,…,K2n−1,Kj)1−p2n−1∏j=1rV(ΠC(Kj[r],Kr+1,…,K2n−1))p2nr.V{\left({\Pi }_{p,C}\left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2n}}\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1}V{\left({K}_{1},\ldots ,{K}_{2n-1},{K}_{j})}^{\tfrac{1-p}{2n-1}}\mathop{\prod }\limits_{j=1}^{r}V{\left({\Pi }_{C}\left({K}_{j}\left[r],{K}_{r+1},\ldots ,{K}_{2n-1}))}^{\tfrac{p}{2nr}}.For p=1p=1, the inequality is Aleksandrov-Fenchel-type inequality for mixed projection bodies [22].In particular, the case of C=[0,1]C=\left[0,1]and λ=0\lambda =0of Theorem 1.3 is Aleksandrov-Fenchel-type inequality for the general Lp{L}_{p}mixed projection bodies [27]. We are now ready to prove Theorem 1.4. Note that Theorem 1.3 reduces to Theorem 1.4 by setting K2n−i=⋯=K2n−1=B{K}_{2n-i}=\cdots ={K}_{2n-1}=B.Proof of Theorem 1.4Recall that M≔(K1,K2,…,K2n−1−i){\bf{M}}:= \left({K}_{1},{K}_{2},\ldots ,{K}_{2n-1-i}). From (2.12), (2.13), and the Hölder’s integral inequality, we have (3.17)h(Πp,i,CλM,u)p≥(2n)1−p∏j=12n−1−iWi(M,Kj)1−p2n−1−ih(Πi,CM,u)p.h{\left({\Pi }_{p,i,C}^{\lambda }{\bf{M}},u)}^{p}\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{2n-1-i}{W}_{i}{\left({\bf{M}},{K}_{j})}^{\tfrac{1-p}{2n-1-i}}h{\left({\Pi }_{i,C}{\bf{M}},u)}^{p}.For all Q∈Ko(Cn)Q\in {{\mathcal{K}}}_{o}\left({{\mathbb{C}}}^{n}), we integrate both sides of (3.17) for dSp(Q,u){\rm{d}}{S}_{p}\left(Q,u)and obtain Vp(Q,Πp,i,CλM)≥(2n)1−pVp(Q,Πi,CM)∏j=12n−1−iWi(M,Kj)1−p2n−1−i.{V}_{p}\left(Q,{\Pi }_{p,i,C}^{\lambda }{\bf{M}})\ge {\left(2n)}^{1-p}{V}_{p}\left(Q,{\Pi }_{i,C}{\bf{M}})\mathop{\prod }\limits_{j=1}^{2n-1-i}{W}_{i}{\left({\bf{M}},{K}_{j})}^{\tfrac{1-p}{2n-1-i}}.Taking Q=Πp,i,CλMQ={\Pi }_{p,i,C}^{\lambda }{\bf{M}}and using (2.5), we obtain V(Πp,i,CλM)p2n≥(2n)1−p∏j=1rV(Πi,C(Kj[r],Kr+1,…,K2n−1−i))p2nr∏j=12n−1−iWi(M,Kj)1−p2n−1−i.V{\left({\Pi }_{p,i,C}^{\lambda }{\bf{M}})}^{\tfrac{p}{2n}}\ge {\left(2n)}^{1-p}\mathop{\prod }\limits_{j=1}^{r}V{\left({\Pi }_{i,C}\left({K}_{j}\left[r],{K}_{r+1},\ldots ,{K}_{2n-1-i}))}^{\tfrac{p}{2nr}}\mathop{\prod }\limits_{j=1}^{2n-1-i}{W}_{i}{\left({\bf{M}},{K}_{j})}^{\tfrac{1-p}{2n-1-i}}.If Kj{K}_{j}is an ellipsoid centered at the origin or an Hermitian ellipsoid, then the equality holds.□

Journal

Open Mathematicsde Gruyter

Published: Jan 1, 2022

Keywords: the general complex Lp mixed projection body; Brunn-Minkowski-type inequalities; Aleksandrov-Fenchel-type inequalities; 52A20; 52A40

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