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Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas

Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas 1IntroductionLet Kn{{\mathcal{K}}}^{n}denote the set of convex bodies (compact, convex subsets with nonempty interiors) in the Euclidean nn-space Rn{{\mathbb{R}}}^{n}. For the set of convex bodies containing the origin in their interiors and the set of convex bodies whose centroid lie at the origin, we write Kon{{\mathcal{K}}}_{o}^{n}and Kcn{{\mathcal{K}}}_{c}^{n}, respectively. For the set of star bodies (about the origin) in Rn{{\mathbb{R}}}^{n}, we write Son{{\mathcal{S}}}_{o}^{n}. Let Sn−1{S}^{n-1}denote the unit sphere in Rn{{\mathbb{R}}}^{n}and V(K)V\left(K)denote the nn-dimensional volume of a body KK. For the standard unit ball BBin Rn{{\mathbb{R}}}^{n}, its volume is written by ωn=V(B){\omega }_{n}=V\left(B).If K∈KnK\in {{\mathcal{K}}}^{n}, then the support function of KK, hK=h(K,⋅):Rn→R,{h}_{K}=h\left(K,\cdot ):{{\mathbb{R}}}^{n}\to {\mathbb{R}},is defined by [3]h(K,x)=max{x⋅y:y∈K},x∈Rn,h\left(K,x)=\max \left\{x\cdot y:y\in K\right\},\hspace{1em}x\in {{\mathbb{R}}}^{n},where x⋅yx\cdot ydenotes the standard inner product of xxand yyin Rn{{\mathbb{R}}}^{n}.Let KKbe a compact star shaped (about the origin) in Rn{{\mathbb{R}}}^{n}, the radial function of KK, ρK=ρ(K,⋅):Rn⧹{0}→[0,+∞){\rho }_{K}=\rho \left(K,\cdot ):{{\mathbb{R}}}^{n}\setminus \left\{0\right\}\to {[}0,+\infty ), is defined by [4]ρ(K,x)=max{λ≥0:λx∈K},x∈Rn⧹{0}.\rho \left(K,x)=\max \left\{\lambda \ge 0:\lambda x\in K\right\},\hspace{1em}x\in {{\mathbb{R}}}^{n}\setminus \left\{0\right\}.If ρK{\rho }_{K}is positive and continuous, KKwill be called a star body (with respect to the origin). Two star bodies KKand LLare dilated (of one another) if ρK(u)/ρL(u){\rho }_{K}\left(u)\hspace{0.1em}\text{/}\hspace{0.1em}{\rho }_{L}\left(u)is independent of u∈Sn−1u\in {S}^{n-1}.For p≥1p\ge 1, the Lp{L}_{p}mixed volume, Vp(K,L){V}_{p}\left(K,L), of K,L∈KonK,L\in {{\mathcal{K}}}_{o}^{n}is defined by [5](1.1)Vp(K,L)=1n∫Sn−1hLp(v)dSp(K,v),{V}_{p}\left(K,L)=\frac{1}{n}\mathop{\int }\limits_{{S}^{n-1}}{h}_{L}^{p}\left(v){\rm{d}}{S}_{p}\left(K,v),where Sp(K,⋅){S}_{p}\left(K,\cdot )is the Lp{L}_{p}surface area measure.Based on the Lp{L}_{p}mixed volume (1.1), Lutwak [6] in 1996 introduced the notion of Lp{L}_{p}geominimal surface areas: For p≥1p\ge 1and K∈KonK\in {{\mathcal{K}}}_{o}^{n}, the Lp{L}_{p}geominimal surface area, Gp(K){G}_{p}\left(K), is defined by ωnpnGp(K)=inf{nVp(K,Q)V(Q∗)pn:Q∈Kon},{\omega }_{n}^{\frac{p}{n}}{G}_{p}\left(K)={\rm{\inf }}\left\{n{V}_{p}\left(K,Q)V{\left({Q}^{\ast })}^{\tfrac{p}{n}}:Q\in {{\mathcal{K}}}_{o}^{n}\right\},where Q∗{Q}^{\ast }denotes the polar of QQ. When p=1p=1, G1(K){G}_{1}\left(K)is just Petty’s classical geominimal surface area [7]. In particular, Lutwak [6] proved the following inequalities.Theorem 1.AIf K∈KonK\in {{\mathcal{K}}}_{o}^{n}and 1≤p<q1\le p\lt q, thenGp(K)nnnV(K)n−p1p≤Gq(K)nnnV(K)n−q1q,{\left(\frac{{G}_{p}{\left(K)}^{n}}{{n}^{n}V{\left(K)}^{n-p}}\right)}^{\tfrac{1}{p}}\le {\left(\frac{{G}_{q}{\left(K)}^{n}}{{n}^{n}V{\left(K)}^{n-q}}\right)}^{\tfrac{1}{q}},with equality if and only if K is p-self-minimal.Theorem 1.BIf K∈KonK\in {{\mathcal{K}}}_{o}^{n}and p≥1p\ge 1, thenωnGp(K)nnnV(K)n−p1p≤V(K)V(K∗),\hspace{-16em}{\omega }_{n}{\left(\frac{{G}_{p}{\left(K)}^{n}}{{n}^{n}V{\left(K)}^{n-p}}\right)}^{\tfrac{1}{p}}\le V\left(K)V\left({K}^{\ast }),with equality if and only if K is p-self-minimal.Very recently, Lutwak et al. [1] introduced the Lp{L}_{p}dual curvature measures as follows: Suppose p,q∈Rp,q\in {\mathbb{R}}. If K∈KonK\in {{\mathcal{K}}}_{o}^{n}while L∈SonL\in {{\mathcal{S}}}_{o}^{n}, then the Lp{L}_{p}dual curvature measures, C˜p,q(K,L,⋅){\widetilde{C}}_{p,q}\left(K,L,\cdot ), on Sn−1{S}^{n-1}are defined by ∫Sn−1g(v)dC˜p,q(K,L,v)=1n∫Sn−1g(αK(u))hK(αK(u))−pρK(u)qρL(u)n−qdu,\mathop{\int }\limits_{{S}^{n-1}}g\left(v){\rm{d}}{\widetilde{C}}_{p,q}\left(K,L,v)=\frac{1}{n}\mathop{\int }\limits_{{S}^{n-1}}g\left({\alpha }_{K}\left(u)){h}_{K}{\left({\alpha }_{K}\left(u))}^{-p}{\rho }_{K}{\left(u)}^{q}{\rho }_{L}{\left(u)}^{n-q}{\rm{d}}u,for each continuous g:Sn−1→Rg:{S}^{n-1}\to {\mathbb{R}}, where αK{\alpha }_{K}is the radial Gauss map (see [1], Section 3).Using the Lp{L}_{p}dual curvature measures, Lutwak et al. [1] defined the (p,q)\left(p,q)-mixed volumes as follows: For p,q∈Rp,q\in {\mathbb{R}}, K,Q∈KonK,Q\in {{\mathcal{K}}}_{o}^{n}and L∈SonL\in {{\mathcal{S}}}_{o}^{n}, the (p,q)\left(p,q)-mixed volume, V˜p,q(K,Q,L){\widetilde{V}}_{p,q}\left(K,Q,L), is defined by V˜p,q(K,Q,L)=∫Sn−1hQp(v)dC˜p,q(K,L,v).{\widetilde{V}}_{p,q}\left(K,Q,L)=\mathop{\int }\limits_{{S}^{n-1}}{h}_{Q}^{p}\left(v){\rm{d}}{\widetilde{C}}_{p,q}\left(K,L,v).Meanwhile, Lutwak et al. [1] gave the following formula of (p,q)\left(p,q)-mixed volume: (1.2)V˜p,q(K,Q,L)=1n∫Sn−1hQhK(αK(u))pρK(u)qρL(u)n−qdu.{\widetilde{V}}_{p,q}\left(K,Q,L)=\frac{1}{n}\mathop{\int }\limits_{{S}^{n-1}}\left(\frac{{h}_{Q}}{{h}_{K}}\right){\left({\alpha }_{K}\left(u))}^{p}{\rho }_{K}{\left(u)}^{q}{\rho }_{L}{\left(u)}^{n-q}{\rm{d}}u.In [1], Lutwak et al. introduced the Lp{L}_{p}mixed volume Vp(K,Q){V}_{p}\left(K,Q)of K,Q∈KonK,Q\in {{\mathcal{K}}}_{o}^{n}for all p∈Rp\in {\mathbb{R}}by Vp(K,Q)=1n∫Sn−1hQp(v)dSp(K,v).{V}_{p}\left(K,Q)=\frac{1}{n}\mathop{\int }\limits_{{S}^{n-1}}{h}_{Q}^{p}\left(v){\rm{d}}{S}_{p}\left(K,v).The case p≥1p\ge 1is just Lutwak’s Lp{L}_{p}mixed volume (1.1). Moreover, for q∈Rq\in {\mathbb{R}}and K,Q∈SonK,Q\in {{\mathcal{S}}}_{o}^{n}, they in [1] also defined the qth dual mixed volume, V˜q(K,Q){\widetilde{V}}_{q}\left(K,Q), by V˜q(K,Q)=1n∫Sn−1ρKq(v)ρQn−q(v)dv.{\widetilde{V}}_{q}\left(K,Q)=\frac{1}{n}\mathop{\int }\limits_{{S}^{n-1}}{\rho }_{K}^{q}\left(v){\rho }_{Q}^{n-q}\left(v){\rm{d}}v.By (1.2), the following special cases are showed: (1.3)V˜p,q(K,Q,K)=Vp(K,Q),\hspace{-13.75em}{\widetilde{V}}_{p,q}\left(K,Q,K)={V}_{p}\left(K,Q),(1.4)V˜p,q(K,K,L)=V˜q(K,L),\hspace{-13.75em}{\widetilde{V}}_{p,q}\left(K,K,L)={\widetilde{V}}_{q}\left(K,L),(1.5)V˜p,n(K,Q,L)=Vp(K,Q).\hspace{-13.75em}{\widetilde{V}}_{p,n}\left(K,Q,L)={V}_{p}\left(K,Q).The (p,q)\left(p,q)-mixed volumes unify the mixed volumes of convex bodies in the Lp{L}_{p}Brunn-Minkowski theory and the dual mixed volumes of star bodies in the Lp{L}_{p}dual Brunn-Minkowski theory. In the last 20 years, the Lp{L}_{p}Brunn-Minkowski theory and its dual theory have been developed very rapidly, see e.g., [3, 4,5,6,8, 9,10,11, 12,13,14, 15,16,17, 18,19,20, 21,22,23].Based on the (p,q)\left(p,q)-mixed volumes, Feng and He in [2] introduced the concept of (p,q)\left(p,q)-mixed geominimal surface areas as follows:Definition 1.1For p,q∈Rp,q\in {\mathbb{R}}, K∈KonK\in {{\mathcal{K}}}_{o}^{n}and L∈SonL\in {{\mathcal{S}}}_{o}^{n}, the (p,q)\left(p,q)-mixed geominimal surface areas, G˜p,q(K,L){\widetilde{G}}_{p,q}\left(K,L), of KKand LLare defined by (1.6)ωnpnG˜p,q(K,L)=inf{nV˜p,q(K,Q,L)V(Q∗)pn:Q∈Kon}.\hspace{-25.45em}{\omega }_{n}^{\frac{p}{n}}{\widetilde{G}}_{p,q}\left(K,L)={\rm{\inf }}\left\{n{\widetilde{V}}_{p,q}\left(K,Q,L)V{\left({Q}^{\ast })}^{\tfrac{p}{n}}:Q\in {{\mathcal{K}}}_{o}^{n}\right\}.If L=KL=Kor q=nq=nin (1.6), then from (1.3) or (1.5) we see that for p≥1p\ge 1the definition is just Lutwak’s Lp{L}_{p}geominimal surface area in [6]. For the studies of Lp{L}_{p}geominimal surface areas, some results have been obtained in these articles (see, e.g., [24,25,26, 27,28,29, 30,31,32, 33,34,35]).According to the definition of (p,q)\left(p,q)-mixed geominimal surface areas, Feng and He [2] extended Theorem 1.A to the following result:Theorem 1.CSuppose q∈Rq\in {\mathbb{R}}. If K∈KonK\in {{\mathcal{K}}}_{o}^{n}and L∈SonL\in {{\mathcal{S}}}_{o}^{n}, then for 0<r<s0\lt r\lt s, G˜r,q(K,L)nnnV˜q(K,L)n−r1r≤G˜s,q(K,L)nnnV˜q(K,L)n−s1s,{\left(\frac{{\widetilde{G}}_{r,q}{\left(K,L)}^{n}}{{n}^{n}{\widetilde{V}}_{q}{\left(K,L)}^{n-r}}\right)}^{\tfrac{1}{r}}\le {\left(\frac{{\widetilde{G}}_{s,q}{\left(K,L)}^{n}}{{n}^{n}{\widetilde{V}}_{q}{\left(K,L)}^{n-s}}\right)}^{\tfrac{1}{s}},with equality if and only if K and L are dilates.Obviously, Theorem 1.C for L=KL=Kand 1≤r<s1\le r\lt simplies Theorem 1.A.In addition, they [2] gave a Brunn-Minkowski-type inequality for the (p,q)\left(p,q)-mixed geominimal surface areas as follows:Theorem 1.DSuppose p,q∈Rp,q\in {\mathbb{R}}such that 0<n−qq<10\lt \frac{n-q}{q}\lt 1and q≠nq\ne n, and let λ,μ∈R\lambda ,\mu \in {\mathbb{R}}. If K∈KonK\in {{\mathcal{K}}}_{o}^{n}, and L1,L2∈Son{L}_{1},{L}_{2}\in {{\mathcal{S}}}_{o}^{n}, thenG˜p,q(K,λ⋅L1+˜qμ⋅L2)qn−q≥λG˜p,q(K,L1)qn−q+μG˜p,q(K,L2)qn−q,{\widetilde{G}}_{p,q}{\left(K,\lambda \cdot {L}_{1}{\widetilde{+}}_{q}\mu \cdot {L}_{2})}^{\tfrac{q}{n-q}}\ge \lambda {\widetilde{G}}_{p,q}{\left(K,{L}_{1})}^{\tfrac{q}{n-q}}+\mu {\widetilde{G}}_{p,q}{\left(K,{L}_{2})}^{\tfrac{q}{n-q}},with equality if and only if L1{L}_{1}and L2{L}_{2}are dilates.2Main resultsHere, λ⋅L1+˜qμ⋅L2\lambda \cdot {L}_{1}{\widetilde{+}}_{q}\mu \cdot {L}_{2}is the Lq{L}_{q}-radial combination of L1{L}_{1}and L2{L}_{2}(see (2.1)).In this article, associated with the (p,q)\left(p,q)-mixed geominimal surface areas, we first establish two monotonic inequalities as follows:Theorem 1.1Suppose p,q∈Rp,q\in {\mathbb{R}}. If K∈KonK\in {{\mathcal{K}}}_{o}^{n}and L∈SonL\in {{\mathcal{S}}}_{o}^{n}, then for 1≤p<q≤n−11\le p\lt q\le n-1, (1.7)G˜p,n−p(K,L)nnnV(K)n−p1p≤G˜q,n−q(K,L)nnnV(K)n−q1q,{\left(\frac{{\widetilde{G}}_{p,n-p}{\left(K,L)}^{n}}{{n}^{n}V{\left(K)}^{n-p}}\right)}^{\tfrac{1}{p}}\le {\left(\frac{{\widetilde{G}}_{q,n-q}{\left(K,L)}^{n}}{{n}^{n}V{\left(K)}^{n-q}}\right)}^{\tfrac{1}{q}},with equality if and only if K and L are dilates.Theorem 1.2Suppose p,q∈Rp,q\in {\mathbb{R}}such that 1≤p<q1\le p\lt q. If K∈KonK\in {{\mathcal{K}}}_{o}^{n}and L∈SonL\in {{\mathcal{S}}}_{o}^{n}, then(1.8)G˜p,p(K,L)nnnV(L)n−p1p≤G˜q,q(K,L)nnnV(L)n−q1q,{\left(\frac{{\widetilde{G}}_{p,p}{\left(K,L)}^{n}}{{n}^{n}V{\left(L)}^{n-p}}\right)}^{\tfrac{1}{p}}\le {\left(\frac{{\widetilde{G}}_{q,q}{\left(K,L)}^{n}}{{n}^{n}V{\left(L)}^{n-q}}\right)}^{\tfrac{1}{q}},with equality if and only if K and L are dilates.Remark 1.1When L=KL=Kand KKis pp-self-minimal, Theorems 1.1 and 1.2 both become Theorem 1.A.Next, we obtain an affine isoperimetric inequality for the (p,q)\left(p,q)-mixed geominimal surface areas.Theorem 1.3Suppose p,q∈Rp,q\in {\mathbb{R}}such that p>0p\gt 0and 0<q≤n0\lt q\le n. If K∈KonK\in {{\mathcal{K}}}_{o}^{n}and L∈SonL\in {{\mathcal{S}}}_{o}^{n}, then(1.9)ωnG˜p,q(K,L)nnnV(K)q−pV(L)n−q1p≤V(K)V(K∗),\hspace{-24.7em}{\omega }_{n}{\left(\frac{{\widetilde{G}}_{p,q}{\left(K,L)}^{n}}{{n}^{n}V{\left(K)}^{q-p}V{\left(L)}^{n-q}}\right)}^{\tfrac{1}{p}}\le V\left(K)V\left({K}^{\ast }),for 0<q<n0\lt q\lt nequality holds when K and L are dilates.Remark 1.2If q=nq=nand p≥1p\ge 1, Theorem 1.3 only contains Lutwak’s inequality.Finally, together with the Lq{L}_{q}harmonic Blaschke combination, we give the following Brunn-Minkowski-type inequality for the (p,q)\left(p,q)-mixed geominimal surface areas.Theorem 1.4Suppose p,q∈Rp,q\in {\mathbb{R}}such that 0<q<n0\lt q\lt n, and let λ,μ≥0\lambda ,\mu \ge 0(not both zero). If K∈KonK\in {{\mathcal{K}}}_{o}^{n}and L1,L2∈Son{L}_{1},{L}_{2}\in {{\mathcal{S}}}_{o}^{n}, then(1.10)G˜p,q(K,λ∗L1+^qμ∗L2)n+qn−qV(λ∗L1+^qμ∗L2)≥λG˜p,q(K,L1)n+qn−qV(L1)+μG˜p,q(K,L2)n+qn−qV(L2),\frac{{\widetilde{G}}_{p,q}{\left(K,\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}^{\tfrac{n+q}{n-q}}}{V\left(\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}\ge \lambda \frac{{\widetilde{G}}_{p,q}{\left(K,{L}_{1})}^{\tfrac{n+q}{n-q}}}{V\left({L}_{1})}+\mu \frac{{\widetilde{G}}_{p,q}{\left(K,{L}_{2})}^{\tfrac{n+q}{n-q}}}{V\left({L}_{2})},with equality if and only if L1{L}_{1}and L2{L}_{2}are dilates.Here, λ∗L1+^qμ∗L2\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2}is the Lq{L}_{q}harmonic Blaschke combination of L1,L2∈Son{L}_{1},{L}_{2}\in {{\mathcal{S}}}_{o}^{n}(see (2.2)).3BackgroundIn order to complete the proofs of Theorems 1.1–1.4, we collect some basic facts about convex bodies and star bodies.If EEis a nonempty subset in Rn{{\mathbb{R}}}^{n}, then the polar set, E∗{E}^{\ast }, of EEis defined by [1,2]E∗={x∈Rn:x⋅y≤1,y∈E}.{E}^{\ast }=\left\{x\in {{\mathbb{R}}}^{n}:x\cdot y\le 1,\hspace{1em}y\in E\right\}.Meanwhile, it is easy to get that (K∗)∗=K{\left({K}^{\ast })}^{\ast }=Kfor all K∈KonK\in {{\mathcal{K}}}_{o}^{n}.For K,L∈SonK,L\in {{\mathcal{S}}}_{o}^{n}, q≠0q\ne 0and λ,μ≥0\lambda ,\mu \ge 0(not both zero), the Lq{L}_{q}radial combination, λ⋅K+˜qμ⋅L∈Son\lambda \cdot K{\tilde{+}}_{q}\mu \cdot L\in {{\mathcal{S}}}_{o}^{n}, of KKand LLis defined by [1](2.1)ρ(λ⋅K+˜qμ⋅L,⋅)q=λρ(K,⋅)q+μρ(L,⋅)q,\hspace{-19.9em}\rho {\left(\lambda \cdot K{\tilde{+}}_{q}\mu \cdot L,\cdot )}^{q}=\lambda \rho {\left(K,\cdot )}^{q}+\mu \rho {\left(L,\cdot )}^{q},where the operation “+˜q{\tilde{+}}_{q}” is called Lq{L}_{q}radial addition, λ⋅K\lambda \cdot Kdenotes Lq{L}_{q}radial scalar multiplication and λ⋅K=λ1qK\lambda \cdot K={\lambda }^{\tfrac{1}{q}}K.For K,L∈SonK,L\in {{\mathcal{S}}}_{o}^{n}, q>0q\gt 0and λ,μ≥0\lambda ,\mu \ge 0(not both zero), the Lq{L}_{q}harmonic Blaschke combination, λ∗K+^qμ∗L∈Son\lambda \ast K{\widehat{+}}_{q}\mu \ast L\in {{\mathcal{S}}}_{o}^{n}, of KKand LLis defined by [36](2.2)ρ(λ∗K+^qμ∗L,⋅)n+qV(λ∗K+^qμ∗L)=λρ(K,⋅)n+qV(K)+μρ(L,⋅)n+qV(L),\hspace{-23.8em}\frac{\rho {\left(\lambda \ast K{\widehat{+}}_{q}\mu \ast L,\cdot )}^{n+q}}{V\left(\lambda \ast K{\widehat{+}}_{q}\mu \ast L)}=\lambda \frac{\rho {\left(K,\cdot )}^{n+q}}{V\left(K)}+\mu \frac{\rho {\left(L,\cdot )}^{n+q}}{V\left(L)},where the operation “+^q{\widehat{+}}_{q}” is called Lq{L}_{q}harmonic Blaschke addition, λ∗K\lambda \ast Kdenotes Lq{L}_{q}harmonic Blaschke scalar multiplication and λ∗K=λ1qK\lambda \ast K={\lambda }^{\tfrac{1}{q}}K. When λ=μ=1\lambda =\mu =1, K+^qLK{\widehat{+}}_{q}Lis called Lq{L}_{q}harmonic Blaschke sum.4Proofs of main theoremsIn this section, we will prove Theorems 1.1–1.4. To complete the proof of Theorem 1.1, we require the following lemma.Lemma 3.1[37] Suppose p,q∈Rp,q\in {\mathbb{R}}. If K,Q∈KonK,Q\in {{\mathcal{K}}}_{o}^{n}and L∈SonL\in {{\mathcal{S}}}_{o}^{n}, then for 0<p<q≤n−10\lt p\lt q\le n-1, (3.1)V˜p,n−p(K,Q,L)V(K)1p≤V˜q,n−q(K,Q,L)V(K)1q,{\left[\frac{{\widetilde{V}}_{p,n-p}\left(K,Q,L)}{V\left(K)}\right]}^{\tfrac{1}{p}}\le {\left[\frac{{\widetilde{V}}_{q,n-q}\left(K,Q,L)}{V\left(K)}\right]}^{\tfrac{1}{q}},with equality if and only if K and L are dilates.Proof of Theorem 1.1Together (1.6) with inequality (3.1), we obtain that for 1≤p<q≤n−11\le p\lt q\le n-1, ωnG˜p,n−p(K,L)nnnV(K)n−p1p=infV˜p,n−p(K,Q,L)V(K)npV(K)V(Q∗):Q∈Kon≤infV˜q,n−q(K,Q,L)V(K)nqV(K)V(Q∗):Q∈Kon=ωnG˜q,n−q(K,L)nnnV(K)n−q1q.\begin{array}{rcl}{\omega }_{n}{\left(\frac{{\widetilde{G}}_{p,n-p}{\left(K,L)}^{n}}{{n}^{n}V{\left(K)}^{n-p}}\right)}^{\tfrac{1}{p}}& =& \inf \left\{{\left(\frac{{\widetilde{V}}_{p,n-p}\left(K,Q,L)}{V\left(K)}\right)}^{\tfrac{n}{p}}V\left(K)V\left({Q}^{\ast }):Q\in {{\mathcal{K}}}_{o}^{n}\right\}\\ & \le & \inf \left\{{\left(\frac{{\widetilde{V}}_{q,n-q}\left(K,Q,L)}{V\left(K)}\right)}^{\tfrac{n}{q}}V\left(K)V\left({Q}^{\ast }):Q\in {{\mathcal{K}}}_{o}^{n}\right\}\\ & =& {\omega }_{n}{\left(\frac{{\widetilde{G}}_{q,n-q}{\left(K,L)}^{n}}{{n}^{n}V{\left(K)}^{n-q}}\right)}^{\tfrac{1}{q}}.\end{array}This is G˜p,n−p(K,L)nnnV(K)n−p1p≤G˜q,n−q(K,L)nnnV(K)n−q1q.{\left(\frac{{\widetilde{G}}_{p,n-p}{\left(K,L)}^{n}}{{n}^{n}V{\left(K)}^{n-p}}\right)}^{\tfrac{1}{p}}\le {\left(\frac{{\widetilde{G}}_{q,n-q}{\left(K,L)}^{n}}{{n}^{n}V{\left(K)}^{n-q}}\right)}^{\tfrac{1}{q}}.This yields inequality (1.7). According to the equality condition of inequality (3.1), we see that the equality of the above inequality holds if and only if KKand LLare dilates.□Lemma 3.2[37] Suppose p,q∈Rp,q\in {\mathbb{R}}satisfy 1≤p<q1\le p\lt q. If K,Q∈KonK,Q\in {{\mathcal{K}}}_{o}^{n}and L∈SonL\in {{\mathcal{S}}}_{o}^{n}, then(3.2)V˜p,p(K,Q,L)V(L)1p≤V˜q,q(K,Q,L)V(L)1q,{\left[\frac{{\widetilde{V}}_{p,p}\left(K,Q,L)}{V\left(L)}\right]}^{\tfrac{1}{p}}\le {\left[\frac{{\widetilde{V}}_{q,q}\left(K,Q,L)}{V\left(L)}\right]}^{\tfrac{1}{q}},with equality if and only if K and L are dilates.Proof of Theorem 1.2From Definition 1.1 and (3.2) we see that for 1≤p<q1\le p\lt q, ωnG˜p,p(K,L)nnnV(L)n−p1p=infV˜p,p(K,Q,L)V(L)npV(L)V(Q∗):Q∈Kon≤infV˜q,q(K,Q,L)V(L)nqV(L)V(Q∗):Q∈Kon=ωnG˜q,q(K,L)nnnV(L)n−q1q.\begin{array}{rcl}{\omega }_{n}{\left(\frac{{\widetilde{G}}_{p,p}{\left(K,L)}^{n}}{{n}^{n}V{\left(L)}^{n-p}}\right)}^{\tfrac{1}{p}}& =& \inf \left\{{\left(\frac{{\widetilde{V}}_{p,p}\left(K,Q,L)}{V\left(L)}\right)}^{\tfrac{n}{p}}V\left(L)V\left({Q}^{\ast }):Q\in {{\mathcal{K}}}_{o}^{n}\right\}\\ & \le & \inf \left\{{\left(\frac{{\widetilde{V}}_{q,q}\left(K,Q,L)}{V\left(L)}\right)}^{\tfrac{n}{q}}V\left(L)V\left({Q}^{\ast }):Q\in {{\mathcal{K}}}_{o}^{n}\right\}\\ & =& {\omega }_{n}{\left(\frac{{\widetilde{G}}_{q,q}{\left(K,L)}^{n}}{{n}^{n}V{\left(L)}^{n-q}}\right)}^{\tfrac{1}{q}}.\end{array}This is, for 1≤p<q1\le p\lt q, G˜p,p(K,L)nnnV(L)n−p1p≤G˜q,q(K,L)nnnV(L)n−q1q.{\left(\frac{{\widetilde{G}}_{p,p}{\left(K,L)}^{n}}{{n}^{n}V{\left(L)}^{n-p}}\right)}^{\tfrac{1}{p}}\le {\left(\frac{{\widetilde{G}}_{q,q}{\left(K,L)}^{n}}{{n}^{n}V{\left(L)}^{n-q}}\right)}^{\tfrac{1}{q}}.This gives inequality (1.8). From the equality condition of inequality (3.2), we see that the equality holds in (1.8) if and only if K,LK,Lare dilates.□Lemma 3.3[2] If K,L∈SonK,L\in {{\mathcal{S}}}_{o}^{n}, 0<q≤n0\lt q\le n, then(3.3)V˜q(K,L)≤V(K)qnV(L)n−qn.{\widetilde{V}}_{q}\left(K,L)\le V{\left(K)}^{\tfrac{q}{n}}V{\left(L)}^{\tfrac{n-q}{n}}.For 0<q<n0\lt q\lt n, equality holds in (3.3) if and only if K and L are dilates; for q=nq=n, (3.3) becomes an equality.Proof of Theorem 1.3Since p>0p\gt 0, it follows from (1.6) that (3.4)ωnG˜p,q(K,L)np=inf{nnpV˜p,q(K,Q,L)npV(Q∗):Q∈Kon}.{\omega }_{n}{\widetilde{G}}_{p,q}{\left(K,L)}^{\tfrac{n}{p}}={\rm{\inf }}\left\{{n}^{\tfrac{n}{p}}{\widetilde{V}}_{p,q}{\left(K,Q,L)}^{\tfrac{n}{p}}V\left({Q}^{\ast }):Q\in {{\mathcal{K}}}_{o}^{n}\right\}.Taking Q=KQ=Kin (3.4), it follows from (1.4) and (3.3) that for 0<q≤n0\lt q\le n, ωnG˜p,q(K,L)np≤nnpV˜p,q(K,K,L)npV(K∗)=nnpV˜q(K,L)npV(K∗)≤nnpV(K)qpV(L)n−qpV(K∗),{\omega }_{n}{\widetilde{G}}_{p,q}{\left(K,L)}^{\tfrac{n}{p}}\le {n}^{\tfrac{n}{p}}{\widetilde{V}}_{p,q}{\left(K,K,L)}^{\tfrac{n}{p}}V\left({K}^{\ast })={n}^{\tfrac{n}{p}}{\widetilde{V}}_{q}{\left(K,L)}^{\tfrac{n}{p}}V\left({K}^{\ast })\le {n}^{\tfrac{n}{p}}V{\left(K)}^{\tfrac{q}{p}}V{\left(L)}^{\tfrac{n-q}{p}}V\left({K}^{\ast }),i.e., ωnG˜p,q(K,L)nnnV(K)q−pV(L)n−q1p≤V(K)V(K∗).\hspace{-24.85em}{\omega }_{n}{\left(\frac{{\widetilde{G}}_{p,q}{\left(K,L)}^{n}}{{n}^{n}V{\left(K)}^{q-p}V{\left(L)}^{n-q}}\right)}^{\tfrac{1}{p}}\le V\left(K)V\left({K}^{\ast }).By the equality condition of inequality (3.3) and Definition 1.1, for 0<q<n0\lt q\lt nequality holds in (1.9) when KKand LLare dilates.□Lemma 3.4Suppose p,q∈Rp,q\in {\mathbb{R}}such that 0<q<n0\lt q\lt n, and let λ,μ>0\lambda ,\mu \gt 0. If K,Q∈KonK,Q\in {{\mathcal{K}}}_{o}^{n}and L1,L2∈Son{L}_{1},{L}_{2}\in {{\mathcal{S}}}_{o}^{n}, then(3.5)V˜p,q(K,Q,λ∗L1+^qμ∗L2)n+qn−qV(λ∗L1+^qμ∗L2)≥λV˜p,q(K,Q,L1)n+qn−qV(L1)+μV˜p,q(K,Q,L2)n+qn−qV(L2),\hspace{1em}\frac{{\widetilde{V}}_{p,q}{\left(K,Q,\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}^{\tfrac{n+q}{n-q}}}{V\left(\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}\ge \lambda \frac{{\widetilde{V}}_{p,q}{\left(K,Q,{L}_{1})}^{\tfrac{n+q}{n-q}}}{V\left({L}_{1})}+\mu \frac{{\widetilde{V}}_{p,q}{\left(K,Q,{L}_{2})}^{\tfrac{n+q}{n-q}}}{V\left({L}_{2})},with equality if and only if L1{L}_{1}and L2{L}_{2}are dilates.ProofSince 0<q<n0\lt q\lt n, thus 0<n−qn+q<10\lt \frac{n-q}{n+q}\lt 1. From (1.2), (2.2) and Minkowski’s integral inequality [38], we get that for any Q∈KonQ\in {{\mathcal{K}}}_{o}^{n}, V˜p,q(K,Q,λ∗L1+^qμ∗L2)n+qn−qV(λ∗L1+^qμ∗L2)=1n∫Sn−1hQhKp(αK(u))ρKq(u)ρλ∗L1+^qμ∗L2n−q(u)dun+qn−qV(λ∗L1+^qμ∗L2)=1n∫Sn−1hQhKp(n+q)n−q(αK(u))ρK(u)q(n+q)n−qρλ∗L1+^qμ∗L2n+q(u)V(λ∗L1+^qμ∗L2)n−qn+qdun+qn−q=1n∫Sn−1hQhKp(n+q)n−q(αK(u))ρK(u)q(n+q)n−qλρL1n+q(u)V(L1)+μρL2n+q(u)V(L2)n−qn+qdun+qn−q≥λ1n∫Sn−1hQhKp(αK(u))ρKq(u)ρL1n−q(u)dun+qn−qV(L1)+μ1n∫Sn−1hQhKp(αK(u))ρKq(u)ρL2n−q(u)dun+qn−qV(L2)=λV˜p,q(K,Q,L1)n+qn−qV(L1)+μV˜p,q(K,Q,L2)n+qn−qV(L2).\begin{array}{rcl}\frac{{\widetilde{V}}_{p,q}{\left(K,Q,\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}^{\tfrac{n+q}{n-q}}}{V\left(\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}& =& \frac{{\left[\frac{1}{n}{\displaystyle \int }_{{S}^{n-1}}{\left(\frac{{h}_{Q}}{{h}_{K}}\right)}^{p}\left({\alpha }_{K}\left(u)){\rho }_{K}^{q}\left(u){\rho }_{\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2}}^{n-q}\left(u){\rm{d}}u\right]}^{\tfrac{n+q}{n-q}}}{V\left(\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}\\ & =& {\left[\frac{1}{n}\mathop{\displaystyle \int }\limits_{{S}^{n-1}}{\left({\left(\frac{{h}_{Q}}{{h}_{K}}\right)}^{\frac{p\left(n+q)}{n-q}}\left({\alpha }_{K}\left(u)){\rho }_{K}{\left(u)}^{\frac{q\left(n+q)}{n-q}}\frac{{\rho }_{\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2}}^{n+q}\left(u)}{V\left(\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}\right)}^{\frac{n-q}{n+q}}{\rm{d}}u\right]}^{\tfrac{n+q}{n-q}}\\ & =& {\left[\frac{1}{n}\mathop{\displaystyle \int }\limits_{{S}^{n-1}}{\left({\left(\frac{{h}_{Q}}{{h}_{K}}\right)}^{\frac{p\left(n+q)}{n-q}}\left({\alpha }_{K}\left(u)){\rho }_{K}{\left(u)}^{\frac{q\left(n+q)}{n-q}}\left(\lambda \frac{{\rho }_{{L}_{1}}^{n+q}\left(u)}{V\left({L}_{1})}+\mu \frac{{\rho }_{{L}_{2}}^{n+q}\left(u)}{V\left({L}_{2})}\right)\right)}^{\frac{n-q}{n+q}}{\rm{d}}u\right]}^{\tfrac{n+q}{n-q}}\\ & \ge & \lambda \frac{{\left[\frac{1}{n}{\displaystyle \int }_{{S}^{n-1}}{\left(\frac{{h}_{Q}}{{h}_{K}}\right)}^{p}\left({\alpha }_{K}\left(u)){\rho }_{K}^{q}\left(u){\rho }_{{L}_{1}}^{n-q}\left(u){\rm{d}}u\right]}^{\tfrac{n+q}{n-q}}}{V\left({L}_{1})}\\ & & \phantom{\rule[-1.5em]{}{0ex}}+\mu \frac{{\left[\frac{1}{n}{\displaystyle \int }_{{S}^{n-1}}{\left(\frac{{h}_{Q}}{{h}_{K}}\right)}^{p}\left({\alpha }_{K}\left(u)){\rho }_{K}^{q}\left(u){\rho }_{{L}_{2}}^{n-q}\left(u){\rm{d}}u\right]}^{\tfrac{n+q}{n-q}}}{V\left({L}_{2})}\\ & =& \lambda \frac{{\widetilde{V}}_{p,q}{\left(K,Q,{L}_{1})}^{\tfrac{n+q}{n-q}}}{V\left({L}_{1})}+\mu \frac{{\widetilde{V}}_{p,q}{\left(K,Q,{L}_{2})}^{\tfrac{n+q}{n-q}}}{V\left({L}_{2})}.\end{array}This yields inequality (3.5).By the equality condition of Minkowski’s integral inequality, we see that equality holds in (3.5) if and only if L1{L}_{1}and L2{L}_{2}are dilates.□Proof of Theorem 1.4Because of 0<q<n0\lt q\lt n, thus n+qn−q>0\frac{n+q}{n-q}\gt 0. Hence, by (1.6) and (3.5) we have ωnpnG˜p,q(K,λ∗L1+^qμ∗L2)n+qn−qV(λ∗L1+^qμ∗L2)=inf[nV˜p,q(K,Q,λ∗L1+^qμ∗L2)]n+qn−qV(λ∗L1+^qμ∗L2)V(Q∗)pnn+qn−q:Q∈Kon≥λinf[nV˜p,q(K,Q,L1)]n+qn−qV(L1)V(Q∗)pnn+qn−q:Q∈Kon+μinf[nV˜p,q(K,Q,L2)]n+qn−qV(L2)V(Q∗)pnn+qn−q:Q∈Kon=λωnpnG˜p,q(K,L1)n+qn−qV(L1)+μωnpnG˜p,q(K,L2)n+qn−qV(L2),\begin{array}{rcl}\frac{{\left[{\omega }_{n}^{\frac{p}{n}}{\widetilde{G}}_{p,q}\left(K,\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})\right]}^{\tfrac{n+q}{n-q}}}{V\left(\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}& =& \inf \left\{\frac{{\left[n{\widetilde{V}}_{p,q}\left(K,Q,\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})]}^{\tfrac{n+q}{n-q}}}{V\left(\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}{\left[V{\left({Q}^{\ast })}^{\frac{p}{n}}\right]}^{\tfrac{n+q}{n-q}}:Q\in {{\mathcal{K}}}_{o}^{n}\right\}\\ & \ge & \lambda \inf \left\{\frac{{\left[n{\widetilde{V}}_{p,q}\left(K,Q,{L}_{1})]}^{\tfrac{n+q}{n-q}}}{V\left({L}_{1})}{\left[V{\left({Q}^{\ast })}^{\frac{p}{n}}\right]}^{\tfrac{n+q}{n-q}}:Q\in {{\mathcal{K}}}_{o}^{n}\right\}\\ & & +\mu \inf \left\{\frac{{\left[n{\widetilde{V}}_{p,q}\left(K,Q,{L}_{2})]}^{\tfrac{n+q}{n-q}}}{V\left({L}_{2})}{\left[V{\left({Q}^{\ast })}^{\frac{p}{n}}\right]}^{\tfrac{n+q}{n-q}}:Q\in {{\mathcal{K}}}_{o}^{n}\right\}\\ & =& \lambda \frac{{\left[{\omega }_{n}^{\frac{p}{n}}{\widetilde{G}}_{p,q}\left(K,{L}_{1})\right]}^{\tfrac{n+q}{n-q}}}{V\left({L}_{1})}+\mu \frac{{\left[{\omega }_{n}^{\frac{p}{n}}{\widetilde{G}}_{p,q}\left(K,{L}_{2})\right]}^{\tfrac{n+q}{n-q}}}{V\left({L}_{2})},\end{array}i.e., G˜p,q(K,λ∗L1+^qμ∗L2)n+qn−qV(λ∗L1+^qμ∗L2)≥λG˜p,q(K,L1)n+qn−qV(L1)+μG˜p,q(K,L2)n+qn−qV(L2).\frac{{\widetilde{G}}_{p,q}{\left(K,\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}^{\tfrac{n+q}{n-q}}}{V\left(\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}\ge \lambda \frac{{\widetilde{G}}_{p,q}{\left(K,{L}_{1})}^{\tfrac{n+q}{n-q}}}{V\left({L}_{1})}+\mu \frac{{\widetilde{G}}_{p,q}{\left(K,{L}_{2})}^{\tfrac{n+q}{n-q}}}{V\left({L}_{2})}.This gives inequality (1.10).According to the equality condition of inequality (3.5), we see that the equality holds in (1.10) if and only if L1{L}_{1}and L2{L}_{2}are dilates.□ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas

Zhang, Juan; Wang, Weidong; Zhao, Peibiao
Open Mathematics , Volume 20 (1): 8 – Jan 1, 2022
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de Gruyter
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© 2022 Juan Zhang et al., published by De Gruyter
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2391-5455
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10.1515/math-2022-0024
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Abstract

1IntroductionLet Kn{{\mathcal{K}}}^{n}denote the set of convex bodies (compact, convex subsets with nonempty interiors) in the Euclidean nn-space Rn{{\mathbb{R}}}^{n}. For the set of convex bodies containing the origin in their interiors and the set of convex bodies whose centroid lie at the origin, we write Kon{{\mathcal{K}}}_{o}^{n}and Kcn{{\mathcal{K}}}_{c}^{n}, respectively. For the set of star bodies (about the origin) in Rn{{\mathbb{R}}}^{n}, we write Son{{\mathcal{S}}}_{o}^{n}. Let Sn−1{S}^{n-1}denote the unit sphere in Rn{{\mathbb{R}}}^{n}and V(K)V\left(K)denote the nn-dimensional volume of a body KK. For the standard unit ball BBin Rn{{\mathbb{R}}}^{n}, its volume is written by ωn=V(B){\omega }_{n}=V\left(B).If K∈KnK\in {{\mathcal{K}}}^{n}, then the support function of KK, hK=h(K,⋅):Rn→R,{h}_{K}=h\left(K,\cdot ):{{\mathbb{R}}}^{n}\to {\mathbb{R}},is defined by [3]h(K,x)=max{x⋅y:y∈K},x∈Rn,h\left(K,x)=\max \left\{x\cdot y:y\in K\right\},\hspace{1em}x\in {{\mathbb{R}}}^{n},where x⋅yx\cdot ydenotes the standard inner product of xxand yyin Rn{{\mathbb{R}}}^{n}.Let KKbe a compact star shaped (about the origin) in Rn{{\mathbb{R}}}^{n}, the radial function of KK, ρK=ρ(K,⋅):Rn⧹{0}→[0,+∞){\rho }_{K}=\rho \left(K,\cdot ):{{\mathbb{R}}}^{n}\setminus \left\{0\right\}\to {[}0,+\infty ), is defined by [4]ρ(K,x)=max{λ≥0:λx∈K},x∈Rn⧹{0}.\rho \left(K,x)=\max \left\{\lambda \ge 0:\lambda x\in K\right\},\hspace{1em}x\in {{\mathbb{R}}}^{n}\setminus \left\{0\right\}.If ρK{\rho }_{K}is positive and continuous, KKwill be called a star body (with respect to the origin). Two star bodies KKand LLare dilated (of one another) if ρK(u)/ρL(u){\rho }_{K}\left(u)\hspace{0.1em}\text{/}\hspace{0.1em}{\rho }_{L}\left(u)is independent of u∈Sn−1u\in {S}^{n-1}.For p≥1p\ge 1, the Lp{L}_{p}mixed volume, Vp(K,L){V}_{p}\left(K,L), of K,L∈KonK,L\in {{\mathcal{K}}}_{o}^{n}is defined by [5](1.1)Vp(K,L)=1n∫Sn−1hLp(v)dSp(K,v),{V}_{p}\left(K,L)=\frac{1}{n}\mathop{\int }\limits_{{S}^{n-1}}{h}_{L}^{p}\left(v){\rm{d}}{S}_{p}\left(K,v),where Sp(K,⋅){S}_{p}\left(K,\cdot )is the Lp{L}_{p}surface area measure.Based on the Lp{L}_{p}mixed volume (1.1), Lutwak [6] in 1996 introduced the notion of Lp{L}_{p}geominimal surface areas: For p≥1p\ge 1and K∈KonK\in {{\mathcal{K}}}_{o}^{n}, the Lp{L}_{p}geominimal surface area, Gp(K){G}_{p}\left(K), is defined by ωnpnGp(K)=inf{nVp(K,Q)V(Q∗)pn:Q∈Kon},{\omega }_{n}^{\frac{p}{n}}{G}_{p}\left(K)={\rm{\inf }}\left\{n{V}_{p}\left(K,Q)V{\left({Q}^{\ast })}^{\tfrac{p}{n}}:Q\in {{\mathcal{K}}}_{o}^{n}\right\},where Q∗{Q}^{\ast }denotes the polar of QQ. When p=1p=1, G1(K){G}_{1}\left(K)is just Petty’s classical geominimal surface area [7]. In particular, Lutwak [6] proved the following inequalities.Theorem 1.AIf K∈KonK\in {{\mathcal{K}}}_{o}^{n}and 1≤p<q1\le p\lt q, thenGp(K)nnnV(K)n−p1p≤Gq(K)nnnV(K)n−q1q,{\left(\frac{{G}_{p}{\left(K)}^{n}}{{n}^{n}V{\left(K)}^{n-p}}\right)}^{\tfrac{1}{p}}\le {\left(\frac{{G}_{q}{\left(K)}^{n}}{{n}^{n}V{\left(K)}^{n-q}}\right)}^{\tfrac{1}{q}},with equality if and only if K is p-self-minimal.Theorem 1.BIf K∈KonK\in {{\mathcal{K}}}_{o}^{n}and p≥1p\ge 1, thenωnGp(K)nnnV(K)n−p1p≤V(K)V(K∗),\hspace{-16em}{\omega }_{n}{\left(\frac{{G}_{p}{\left(K)}^{n}}{{n}^{n}V{\left(K)}^{n-p}}\right)}^{\tfrac{1}{p}}\le V\left(K)V\left({K}^{\ast }),with equality if and only if K is p-self-minimal.Very recently, Lutwak et al. [1] introduced the Lp{L}_{p}dual curvature measures as follows: Suppose p,q∈Rp,q\in {\mathbb{R}}. If K∈KonK\in {{\mathcal{K}}}_{o}^{n}while L∈SonL\in {{\mathcal{S}}}_{o}^{n}, then the Lp{L}_{p}dual curvature measures, C˜p,q(K,L,⋅){\widetilde{C}}_{p,q}\left(K,L,\cdot ), on Sn−1{S}^{n-1}are defined by ∫Sn−1g(v)dC˜p,q(K,L,v)=1n∫Sn−1g(αK(u))hK(αK(u))−pρK(u)qρL(u)n−qdu,\mathop{\int }\limits_{{S}^{n-1}}g\left(v){\rm{d}}{\widetilde{C}}_{p,q}\left(K,L,v)=\frac{1}{n}\mathop{\int }\limits_{{S}^{n-1}}g\left({\alpha }_{K}\left(u)){h}_{K}{\left({\alpha }_{K}\left(u))}^{-p}{\rho }_{K}{\left(u)}^{q}{\rho }_{L}{\left(u)}^{n-q}{\rm{d}}u,for each continuous g:Sn−1→Rg:{S}^{n-1}\to {\mathbb{R}}, where αK{\alpha }_{K}is the radial Gauss map (see [1], Section 3).Using the Lp{L}_{p}dual curvature measures, Lutwak et al. [1] defined the (p,q)\left(p,q)-mixed volumes as follows: For p,q∈Rp,q\in {\mathbb{R}}, K,Q∈KonK,Q\in {{\mathcal{K}}}_{o}^{n}and L∈SonL\in {{\mathcal{S}}}_{o}^{n}, the (p,q)\left(p,q)-mixed volume, V˜p,q(K,Q,L){\widetilde{V}}_{p,q}\left(K,Q,L), is defined by V˜p,q(K,Q,L)=∫Sn−1hQp(v)dC˜p,q(K,L,v).{\widetilde{V}}_{p,q}\left(K,Q,L)=\mathop{\int }\limits_{{S}^{n-1}}{h}_{Q}^{p}\left(v){\rm{d}}{\widetilde{C}}_{p,q}\left(K,L,v).Meanwhile, Lutwak et al. [1] gave the following formula of (p,q)\left(p,q)-mixed volume: (1.2)V˜p,q(K,Q,L)=1n∫Sn−1hQhK(αK(u))pρK(u)qρL(u)n−qdu.{\widetilde{V}}_{p,q}\left(K,Q,L)=\frac{1}{n}\mathop{\int }\limits_{{S}^{n-1}}\left(\frac{{h}_{Q}}{{h}_{K}}\right){\left({\alpha }_{K}\left(u))}^{p}{\rho }_{K}{\left(u)}^{q}{\rho }_{L}{\left(u)}^{n-q}{\rm{d}}u.In [1], Lutwak et al. introduced the Lp{L}_{p}mixed volume Vp(K,Q){V}_{p}\left(K,Q)of K,Q∈KonK,Q\in {{\mathcal{K}}}_{o}^{n}for all p∈Rp\in {\mathbb{R}}by Vp(K,Q)=1n∫Sn−1hQp(v)dSp(K,v).{V}_{p}\left(K,Q)=\frac{1}{n}\mathop{\int }\limits_{{S}^{n-1}}{h}_{Q}^{p}\left(v){\rm{d}}{S}_{p}\left(K,v).The case p≥1p\ge 1is just Lutwak’s Lp{L}_{p}mixed volume (1.1). Moreover, for q∈Rq\in {\mathbb{R}}and K,Q∈SonK,Q\in {{\mathcal{S}}}_{o}^{n}, they in [1] also defined the qth dual mixed volume, V˜q(K,Q){\widetilde{V}}_{q}\left(K,Q), by V˜q(K,Q)=1n∫Sn−1ρKq(v)ρQn−q(v)dv.{\widetilde{V}}_{q}\left(K,Q)=\frac{1}{n}\mathop{\int }\limits_{{S}^{n-1}}{\rho }_{K}^{q}\left(v){\rho }_{Q}^{n-q}\left(v){\rm{d}}v.By (1.2), the following special cases are showed: (1.3)V˜p,q(K,Q,K)=Vp(K,Q),\hspace{-13.75em}{\widetilde{V}}_{p,q}\left(K,Q,K)={V}_{p}\left(K,Q),(1.4)V˜p,q(K,K,L)=V˜q(K,L),\hspace{-13.75em}{\widetilde{V}}_{p,q}\left(K,K,L)={\widetilde{V}}_{q}\left(K,L),(1.5)V˜p,n(K,Q,L)=Vp(K,Q).\hspace{-13.75em}{\widetilde{V}}_{p,n}\left(K,Q,L)={V}_{p}\left(K,Q).The (p,q)\left(p,q)-mixed volumes unify the mixed volumes of convex bodies in the Lp{L}_{p}Brunn-Minkowski theory and the dual mixed volumes of star bodies in the Lp{L}_{p}dual Brunn-Minkowski theory. In the last 20 years, the Lp{L}_{p}Brunn-Minkowski theory and its dual theory have been developed very rapidly, see e.g., [3, 4,5,6,8, 9,10,11, 12,13,14, 15,16,17, 18,19,20, 21,22,23].Based on the (p,q)\left(p,q)-mixed volumes, Feng and He in [2] introduced the concept of (p,q)\left(p,q)-mixed geominimal surface areas as follows:Definition 1.1For p,q∈Rp,q\in {\mathbb{R}}, K∈KonK\in {{\mathcal{K}}}_{o}^{n}and L∈SonL\in {{\mathcal{S}}}_{o}^{n}, the (p,q)\left(p,q)-mixed geominimal surface areas, G˜p,q(K,L){\widetilde{G}}_{p,q}\left(K,L), of KKand LLare defined by (1.6)ωnpnG˜p,q(K,L)=inf{nV˜p,q(K,Q,L)V(Q∗)pn:Q∈Kon}.\hspace{-25.45em}{\omega }_{n}^{\frac{p}{n}}{\widetilde{G}}_{p,q}\left(K,L)={\rm{\inf }}\left\{n{\widetilde{V}}_{p,q}\left(K,Q,L)V{\left({Q}^{\ast })}^{\tfrac{p}{n}}:Q\in {{\mathcal{K}}}_{o}^{n}\right\}.If L=KL=Kor q=nq=nin (1.6), then from (1.3) or (1.5) we see that for p≥1p\ge 1the definition is just Lutwak’s Lp{L}_{p}geominimal surface area in [6]. For the studies of Lp{L}_{p}geominimal surface areas, some results have been obtained in these articles (see, e.g., [24,25,26, 27,28,29, 30,31,32, 33,34,35]).According to the definition of (p,q)\left(p,q)-mixed geominimal surface areas, Feng and He [2] extended Theorem 1.A to the following result:Theorem 1.CSuppose q∈Rq\in {\mathbb{R}}. If K∈KonK\in {{\mathcal{K}}}_{o}^{n}and L∈SonL\in {{\mathcal{S}}}_{o}^{n}, then for 0<r<s0\lt r\lt s, G˜r,q(K,L)nnnV˜q(K,L)n−r1r≤G˜s,q(K,L)nnnV˜q(K,L)n−s1s,{\left(\frac{{\widetilde{G}}_{r,q}{\left(K,L)}^{n}}{{n}^{n}{\widetilde{V}}_{q}{\left(K,L)}^{n-r}}\right)}^{\tfrac{1}{r}}\le {\left(\frac{{\widetilde{G}}_{s,q}{\left(K,L)}^{n}}{{n}^{n}{\widetilde{V}}_{q}{\left(K,L)}^{n-s}}\right)}^{\tfrac{1}{s}},with equality if and only if K and L are dilates.Obviously, Theorem 1.C for L=KL=Kand 1≤r<s1\le r\lt simplies Theorem 1.A.In addition, they [2] gave a Brunn-Minkowski-type inequality for the (p,q)\left(p,q)-mixed geominimal surface areas as follows:Theorem 1.DSuppose p,q∈Rp,q\in {\mathbb{R}}such that 0<n−qq<10\lt \frac{n-q}{q}\lt 1and q≠nq\ne n, and let λ,μ∈R\lambda ,\mu \in {\mathbb{R}}. If K∈KonK\in {{\mathcal{K}}}_{o}^{n}, and L1,L2∈Son{L}_{1},{L}_{2}\in {{\mathcal{S}}}_{o}^{n}, thenG˜p,q(K,λ⋅L1+˜qμ⋅L2)qn−q≥λG˜p,q(K,L1)qn−q+μG˜p,q(K,L2)qn−q,{\widetilde{G}}_{p,q}{\left(K,\lambda \cdot {L}_{1}{\widetilde{+}}_{q}\mu \cdot {L}_{2})}^{\tfrac{q}{n-q}}\ge \lambda {\widetilde{G}}_{p,q}{\left(K,{L}_{1})}^{\tfrac{q}{n-q}}+\mu {\widetilde{G}}_{p,q}{\left(K,{L}_{2})}^{\tfrac{q}{n-q}},with equality if and only if L1{L}_{1}and L2{L}_{2}are dilates.2Main resultsHere, λ⋅L1+˜qμ⋅L2\lambda \cdot {L}_{1}{\widetilde{+}}_{q}\mu \cdot {L}_{2}is the Lq{L}_{q}-radial combination of L1{L}_{1}and L2{L}_{2}(see (2.1)).In this article, associated with the (p,q)\left(p,q)-mixed geominimal surface areas, we first establish two monotonic inequalities as follows:Theorem 1.1Suppose p,q∈Rp,q\in {\mathbb{R}}. If K∈KonK\in {{\mathcal{K}}}_{o}^{n}and L∈SonL\in {{\mathcal{S}}}_{o}^{n}, then for 1≤p<q≤n−11\le p\lt q\le n-1, (1.7)G˜p,n−p(K,L)nnnV(K)n−p1p≤G˜q,n−q(K,L)nnnV(K)n−q1q,{\left(\frac{{\widetilde{G}}_{p,n-p}{\left(K,L)}^{n}}{{n}^{n}V{\left(K)}^{n-p}}\right)}^{\tfrac{1}{p}}\le {\left(\frac{{\widetilde{G}}_{q,n-q}{\left(K,L)}^{n}}{{n}^{n}V{\left(K)}^{n-q}}\right)}^{\tfrac{1}{q}},with equality if and only if K and L are dilates.Theorem 1.2Suppose p,q∈Rp,q\in {\mathbb{R}}such that 1≤p<q1\le p\lt q. If K∈KonK\in {{\mathcal{K}}}_{o}^{n}and L∈SonL\in {{\mathcal{S}}}_{o}^{n}, then(1.8)G˜p,p(K,L)nnnV(L)n−p1p≤G˜q,q(K,L)nnnV(L)n−q1q,{\left(\frac{{\widetilde{G}}_{p,p}{\left(K,L)}^{n}}{{n}^{n}V{\left(L)}^{n-p}}\right)}^{\tfrac{1}{p}}\le {\left(\frac{{\widetilde{G}}_{q,q}{\left(K,L)}^{n}}{{n}^{n}V{\left(L)}^{n-q}}\right)}^{\tfrac{1}{q}},with equality if and only if K and L are dilates.Remark 1.1When L=KL=Kand KKis pp-self-minimal, Theorems 1.1 and 1.2 both become Theorem 1.A.Next, we obtain an affine isoperimetric inequality for the (p,q)\left(p,q)-mixed geominimal surface areas.Theorem 1.3Suppose p,q∈Rp,q\in {\mathbb{R}}such that p>0p\gt 0and 0<q≤n0\lt q\le n. If K∈KonK\in {{\mathcal{K}}}_{o}^{n}and L∈SonL\in {{\mathcal{S}}}_{o}^{n}, then(1.9)ωnG˜p,q(K,L)nnnV(K)q−pV(L)n−q1p≤V(K)V(K∗),\hspace{-24.7em}{\omega }_{n}{\left(\frac{{\widetilde{G}}_{p,q}{\left(K,L)}^{n}}{{n}^{n}V{\left(K)}^{q-p}V{\left(L)}^{n-q}}\right)}^{\tfrac{1}{p}}\le V\left(K)V\left({K}^{\ast }),for 0<q<n0\lt q\lt nequality holds when K and L are dilates.Remark 1.2If q=nq=nand p≥1p\ge 1, Theorem 1.3 only contains Lutwak’s inequality.Finally, together with the Lq{L}_{q}harmonic Blaschke combination, we give the following Brunn-Minkowski-type inequality for the (p,q)\left(p,q)-mixed geominimal surface areas.Theorem 1.4Suppose p,q∈Rp,q\in {\mathbb{R}}such that 0<q<n0\lt q\lt n, and let λ,μ≥0\lambda ,\mu \ge 0(not both zero). If K∈KonK\in {{\mathcal{K}}}_{o}^{n}and L1,L2∈Son{L}_{1},{L}_{2}\in {{\mathcal{S}}}_{o}^{n}, then(1.10)G˜p,q(K,λ∗L1+^qμ∗L2)n+qn−qV(λ∗L1+^qμ∗L2)≥λG˜p,q(K,L1)n+qn−qV(L1)+μG˜p,q(K,L2)n+qn−qV(L2),\frac{{\widetilde{G}}_{p,q}{\left(K,\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}^{\tfrac{n+q}{n-q}}}{V\left(\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}\ge \lambda \frac{{\widetilde{G}}_{p,q}{\left(K,{L}_{1})}^{\tfrac{n+q}{n-q}}}{V\left({L}_{1})}+\mu \frac{{\widetilde{G}}_{p,q}{\left(K,{L}_{2})}^{\tfrac{n+q}{n-q}}}{V\left({L}_{2})},with equality if and only if L1{L}_{1}and L2{L}_{2}are dilates.Here, λ∗L1+^qμ∗L2\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2}is the Lq{L}_{q}harmonic Blaschke combination of L1,L2∈Son{L}_{1},{L}_{2}\in {{\mathcal{S}}}_{o}^{n}(see (2.2)).3BackgroundIn order to complete the proofs of Theorems 1.1–1.4, we collect some basic facts about convex bodies and star bodies.If EEis a nonempty subset in Rn{{\mathbb{R}}}^{n}, then the polar set, E∗{E}^{\ast }, of EEis defined by [1,2]E∗={x∈Rn:x⋅y≤1,y∈E}.{E}^{\ast }=\left\{x\in {{\mathbb{R}}}^{n}:x\cdot y\le 1,\hspace{1em}y\in E\right\}.Meanwhile, it is easy to get that (K∗)∗=K{\left({K}^{\ast })}^{\ast }=Kfor all K∈KonK\in {{\mathcal{K}}}_{o}^{n}.For K,L∈SonK,L\in {{\mathcal{S}}}_{o}^{n}, q≠0q\ne 0and λ,μ≥0\lambda ,\mu \ge 0(not both zero), the Lq{L}_{q}radial combination, λ⋅K+˜qμ⋅L∈Son\lambda \cdot K{\tilde{+}}_{q}\mu \cdot L\in {{\mathcal{S}}}_{o}^{n}, of KKand LLis defined by [1](2.1)ρ(λ⋅K+˜qμ⋅L,⋅)q=λρ(K,⋅)q+μρ(L,⋅)q,\hspace{-19.9em}\rho {\left(\lambda \cdot K{\tilde{+}}_{q}\mu \cdot L,\cdot )}^{q}=\lambda \rho {\left(K,\cdot )}^{q}+\mu \rho {\left(L,\cdot )}^{q},where the operation “+˜q{\tilde{+}}_{q}” is called Lq{L}_{q}radial addition, λ⋅K\lambda \cdot Kdenotes Lq{L}_{q}radial scalar multiplication and λ⋅K=λ1qK\lambda \cdot K={\lambda }^{\tfrac{1}{q}}K.For K,L∈SonK,L\in {{\mathcal{S}}}_{o}^{n}, q>0q\gt 0and λ,μ≥0\lambda ,\mu \ge 0(not both zero), the Lq{L}_{q}harmonic Blaschke combination, λ∗K+^qμ∗L∈Son\lambda \ast K{\widehat{+}}_{q}\mu \ast L\in {{\mathcal{S}}}_{o}^{n}, of KKand LLis defined by [36](2.2)ρ(λ∗K+^qμ∗L,⋅)n+qV(λ∗K+^qμ∗L)=λρ(K,⋅)n+qV(K)+μρ(L,⋅)n+qV(L),\hspace{-23.8em}\frac{\rho {\left(\lambda \ast K{\widehat{+}}_{q}\mu \ast L,\cdot )}^{n+q}}{V\left(\lambda \ast K{\widehat{+}}_{q}\mu \ast L)}=\lambda \frac{\rho {\left(K,\cdot )}^{n+q}}{V\left(K)}+\mu \frac{\rho {\left(L,\cdot )}^{n+q}}{V\left(L)},where the operation “+^q{\widehat{+}}_{q}” is called Lq{L}_{q}harmonic Blaschke addition, λ∗K\lambda \ast Kdenotes Lq{L}_{q}harmonic Blaschke scalar multiplication and λ∗K=λ1qK\lambda \ast K={\lambda }^{\tfrac{1}{q}}K. When λ=μ=1\lambda =\mu =1, K+^qLK{\widehat{+}}_{q}Lis called Lq{L}_{q}harmonic Blaschke sum.4Proofs of main theoremsIn this section, we will prove Theorems 1.1–1.4. To complete the proof of Theorem 1.1, we require the following lemma.Lemma 3.1[37] Suppose p,q∈Rp,q\in {\mathbb{R}}. If K,Q∈KonK,Q\in {{\mathcal{K}}}_{o}^{n}and L∈SonL\in {{\mathcal{S}}}_{o}^{n}, then for 0<p<q≤n−10\lt p\lt q\le n-1, (3.1)V˜p,n−p(K,Q,L)V(K)1p≤V˜q,n−q(K,Q,L)V(K)1q,{\left[\frac{{\widetilde{V}}_{p,n-p}\left(K,Q,L)}{V\left(K)}\right]}^{\tfrac{1}{p}}\le {\left[\frac{{\widetilde{V}}_{q,n-q}\left(K,Q,L)}{V\left(K)}\right]}^{\tfrac{1}{q}},with equality if and only if K and L are dilates.Proof of Theorem 1.1Together (1.6) with inequality (3.1), we obtain that for 1≤p<q≤n−11\le p\lt q\le n-1, ωnG˜p,n−p(K,L)nnnV(K)n−p1p=infV˜p,n−p(K,Q,L)V(K)npV(K)V(Q∗):Q∈Kon≤infV˜q,n−q(K,Q,L)V(K)nqV(K)V(Q∗):Q∈Kon=ωnG˜q,n−q(K,L)nnnV(K)n−q1q.\begin{array}{rcl}{\omega }_{n}{\left(\frac{{\widetilde{G}}_{p,n-p}{\left(K,L)}^{n}}{{n}^{n}V{\left(K)}^{n-p}}\right)}^{\tfrac{1}{p}}& =& \inf \left\{{\left(\frac{{\widetilde{V}}_{p,n-p}\left(K,Q,L)}{V\left(K)}\right)}^{\tfrac{n}{p}}V\left(K)V\left({Q}^{\ast }):Q\in {{\mathcal{K}}}_{o}^{n}\right\}\\ & \le & \inf \left\{{\left(\frac{{\widetilde{V}}_{q,n-q}\left(K,Q,L)}{V\left(K)}\right)}^{\tfrac{n}{q}}V\left(K)V\left({Q}^{\ast }):Q\in {{\mathcal{K}}}_{o}^{n}\right\}\\ & =& {\omega }_{n}{\left(\frac{{\widetilde{G}}_{q,n-q}{\left(K,L)}^{n}}{{n}^{n}V{\left(K)}^{n-q}}\right)}^{\tfrac{1}{q}}.\end{array}This is G˜p,n−p(K,L)nnnV(K)n−p1p≤G˜q,n−q(K,L)nnnV(K)n−q1q.{\left(\frac{{\widetilde{G}}_{p,n-p}{\left(K,L)}^{n}}{{n}^{n}V{\left(K)}^{n-p}}\right)}^{\tfrac{1}{p}}\le {\left(\frac{{\widetilde{G}}_{q,n-q}{\left(K,L)}^{n}}{{n}^{n}V{\left(K)}^{n-q}}\right)}^{\tfrac{1}{q}}.This yields inequality (1.7). According to the equality condition of inequality (3.1), we see that the equality of the above inequality holds if and only if KKand LLare dilates.□Lemma 3.2[37] Suppose p,q∈Rp,q\in {\mathbb{R}}satisfy 1≤p<q1\le p\lt q. If K,Q∈KonK,Q\in {{\mathcal{K}}}_{o}^{n}and L∈SonL\in {{\mathcal{S}}}_{o}^{n}, then(3.2)V˜p,p(K,Q,L)V(L)1p≤V˜q,q(K,Q,L)V(L)1q,{\left[\frac{{\widetilde{V}}_{p,p}\left(K,Q,L)}{V\left(L)}\right]}^{\tfrac{1}{p}}\le {\left[\frac{{\widetilde{V}}_{q,q}\left(K,Q,L)}{V\left(L)}\right]}^{\tfrac{1}{q}},with equality if and only if K and L are dilates.Proof of Theorem 1.2From Definition 1.1 and (3.2) we see that for 1≤p<q1\le p\lt q, ωnG˜p,p(K,L)nnnV(L)n−p1p=infV˜p,p(K,Q,L)V(L)npV(L)V(Q∗):Q∈Kon≤infV˜q,q(K,Q,L)V(L)nqV(L)V(Q∗):Q∈Kon=ωnG˜q,q(K,L)nnnV(L)n−q1q.\begin{array}{rcl}{\omega }_{n}{\left(\frac{{\widetilde{G}}_{p,p}{\left(K,L)}^{n}}{{n}^{n}V{\left(L)}^{n-p}}\right)}^{\tfrac{1}{p}}& =& \inf \left\{{\left(\frac{{\widetilde{V}}_{p,p}\left(K,Q,L)}{V\left(L)}\right)}^{\tfrac{n}{p}}V\left(L)V\left({Q}^{\ast }):Q\in {{\mathcal{K}}}_{o}^{n}\right\}\\ & \le & \inf \left\{{\left(\frac{{\widetilde{V}}_{q,q}\left(K,Q,L)}{V\left(L)}\right)}^{\tfrac{n}{q}}V\left(L)V\left({Q}^{\ast }):Q\in {{\mathcal{K}}}_{o}^{n}\right\}\\ & =& {\omega }_{n}{\left(\frac{{\widetilde{G}}_{q,q}{\left(K,L)}^{n}}{{n}^{n}V{\left(L)}^{n-q}}\right)}^{\tfrac{1}{q}}.\end{array}This is, for 1≤p<q1\le p\lt q, G˜p,p(K,L)nnnV(L)n−p1p≤G˜q,q(K,L)nnnV(L)n−q1q.{\left(\frac{{\widetilde{G}}_{p,p}{\left(K,L)}^{n}}{{n}^{n}V{\left(L)}^{n-p}}\right)}^{\tfrac{1}{p}}\le {\left(\frac{{\widetilde{G}}_{q,q}{\left(K,L)}^{n}}{{n}^{n}V{\left(L)}^{n-q}}\right)}^{\tfrac{1}{q}}.This gives inequality (1.8). From the equality condition of inequality (3.2), we see that the equality holds in (1.8) if and only if K,LK,Lare dilates.□Lemma 3.3[2] If K,L∈SonK,L\in {{\mathcal{S}}}_{o}^{n}, 0<q≤n0\lt q\le n, then(3.3)V˜q(K,L)≤V(K)qnV(L)n−qn.{\widetilde{V}}_{q}\left(K,L)\le V{\left(K)}^{\tfrac{q}{n}}V{\left(L)}^{\tfrac{n-q}{n}}.For 0<q<n0\lt q\lt n, equality holds in (3.3) if and only if K and L are dilates; for q=nq=n, (3.3) becomes an equality.Proof of Theorem 1.3Since p>0p\gt 0, it follows from (1.6) that (3.4)ωnG˜p,q(K,L)np=inf{nnpV˜p,q(K,Q,L)npV(Q∗):Q∈Kon}.{\omega }_{n}{\widetilde{G}}_{p,q}{\left(K,L)}^{\tfrac{n}{p}}={\rm{\inf }}\left\{{n}^{\tfrac{n}{p}}{\widetilde{V}}_{p,q}{\left(K,Q,L)}^{\tfrac{n}{p}}V\left({Q}^{\ast }):Q\in {{\mathcal{K}}}_{o}^{n}\right\}.Taking Q=KQ=Kin (3.4), it follows from (1.4) and (3.3) that for 0<q≤n0\lt q\le n, ωnG˜p,q(K,L)np≤nnpV˜p,q(K,K,L)npV(K∗)=nnpV˜q(K,L)npV(K∗)≤nnpV(K)qpV(L)n−qpV(K∗),{\omega }_{n}{\widetilde{G}}_{p,q}{\left(K,L)}^{\tfrac{n}{p}}\le {n}^{\tfrac{n}{p}}{\widetilde{V}}_{p,q}{\left(K,K,L)}^{\tfrac{n}{p}}V\left({K}^{\ast })={n}^{\tfrac{n}{p}}{\widetilde{V}}_{q}{\left(K,L)}^{\tfrac{n}{p}}V\left({K}^{\ast })\le {n}^{\tfrac{n}{p}}V{\left(K)}^{\tfrac{q}{p}}V{\left(L)}^{\tfrac{n-q}{p}}V\left({K}^{\ast }),i.e., ωnG˜p,q(K,L)nnnV(K)q−pV(L)n−q1p≤V(K)V(K∗).\hspace{-24.85em}{\omega }_{n}{\left(\frac{{\widetilde{G}}_{p,q}{\left(K,L)}^{n}}{{n}^{n}V{\left(K)}^{q-p}V{\left(L)}^{n-q}}\right)}^{\tfrac{1}{p}}\le V\left(K)V\left({K}^{\ast }).By the equality condition of inequality (3.3) and Definition 1.1, for 0<q<n0\lt q\lt nequality holds in (1.9) when KKand LLare dilates.□Lemma 3.4Suppose p,q∈Rp,q\in {\mathbb{R}}such that 0<q<n0\lt q\lt n, and let λ,μ>0\lambda ,\mu \gt 0. If K,Q∈KonK,Q\in {{\mathcal{K}}}_{o}^{n}and L1,L2∈Son{L}_{1},{L}_{2}\in {{\mathcal{S}}}_{o}^{n}, then(3.5)V˜p,q(K,Q,λ∗L1+^qμ∗L2)n+qn−qV(λ∗L1+^qμ∗L2)≥λV˜p,q(K,Q,L1)n+qn−qV(L1)+μV˜p,q(K,Q,L2)n+qn−qV(L2),\hspace{1em}\frac{{\widetilde{V}}_{p,q}{\left(K,Q,\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}^{\tfrac{n+q}{n-q}}}{V\left(\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}\ge \lambda \frac{{\widetilde{V}}_{p,q}{\left(K,Q,{L}_{1})}^{\tfrac{n+q}{n-q}}}{V\left({L}_{1})}+\mu \frac{{\widetilde{V}}_{p,q}{\left(K,Q,{L}_{2})}^{\tfrac{n+q}{n-q}}}{V\left({L}_{2})},with equality if and only if L1{L}_{1}and L2{L}_{2}are dilates.ProofSince 0<q<n0\lt q\lt n, thus 0<n−qn+q<10\lt \frac{n-q}{n+q}\lt 1. From (1.2), (2.2) and Minkowski’s integral inequality [38], we get that for any Q∈KonQ\in {{\mathcal{K}}}_{o}^{n}, V˜p,q(K,Q,λ∗L1+^qμ∗L2)n+qn−qV(λ∗L1+^qμ∗L2)=1n∫Sn−1hQhKp(αK(u))ρKq(u)ρλ∗L1+^qμ∗L2n−q(u)dun+qn−qV(λ∗L1+^qμ∗L2)=1n∫Sn−1hQhKp(n+q)n−q(αK(u))ρK(u)q(n+q)n−qρλ∗L1+^qμ∗L2n+q(u)V(λ∗L1+^qμ∗L2)n−qn+qdun+qn−q=1n∫Sn−1hQhKp(n+q)n−q(αK(u))ρK(u)q(n+q)n−qλρL1n+q(u)V(L1)+μρL2n+q(u)V(L2)n−qn+qdun+qn−q≥λ1n∫Sn−1hQhKp(αK(u))ρKq(u)ρL1n−q(u)dun+qn−qV(L1)+μ1n∫Sn−1hQhKp(αK(u))ρKq(u)ρL2n−q(u)dun+qn−qV(L2)=λV˜p,q(K,Q,L1)n+qn−qV(L1)+μV˜p,q(K,Q,L2)n+qn−qV(L2).\begin{array}{rcl}\frac{{\widetilde{V}}_{p,q}{\left(K,Q,\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}^{\tfrac{n+q}{n-q}}}{V\left(\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}& =& \frac{{\left[\frac{1}{n}{\displaystyle \int }_{{S}^{n-1}}{\left(\frac{{h}_{Q}}{{h}_{K}}\right)}^{p}\left({\alpha }_{K}\left(u)){\rho }_{K}^{q}\left(u){\rho }_{\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2}}^{n-q}\left(u){\rm{d}}u\right]}^{\tfrac{n+q}{n-q}}}{V\left(\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}\\ & =& {\left[\frac{1}{n}\mathop{\displaystyle \int }\limits_{{S}^{n-1}}{\left({\left(\frac{{h}_{Q}}{{h}_{K}}\right)}^{\frac{p\left(n+q)}{n-q}}\left({\alpha }_{K}\left(u)){\rho }_{K}{\left(u)}^{\frac{q\left(n+q)}{n-q}}\frac{{\rho }_{\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2}}^{n+q}\left(u)}{V\left(\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}\right)}^{\frac{n-q}{n+q}}{\rm{d}}u\right]}^{\tfrac{n+q}{n-q}}\\ & =& {\left[\frac{1}{n}\mathop{\displaystyle \int }\limits_{{S}^{n-1}}{\left({\left(\frac{{h}_{Q}}{{h}_{K}}\right)}^{\frac{p\left(n+q)}{n-q}}\left({\alpha }_{K}\left(u)){\rho }_{K}{\left(u)}^{\frac{q\left(n+q)}{n-q}}\left(\lambda \frac{{\rho }_{{L}_{1}}^{n+q}\left(u)}{V\left({L}_{1})}+\mu \frac{{\rho }_{{L}_{2}}^{n+q}\left(u)}{V\left({L}_{2})}\right)\right)}^{\frac{n-q}{n+q}}{\rm{d}}u\right]}^{\tfrac{n+q}{n-q}}\\ & \ge & \lambda \frac{{\left[\frac{1}{n}{\displaystyle \int }_{{S}^{n-1}}{\left(\frac{{h}_{Q}}{{h}_{K}}\right)}^{p}\left({\alpha }_{K}\left(u)){\rho }_{K}^{q}\left(u){\rho }_{{L}_{1}}^{n-q}\left(u){\rm{d}}u\right]}^{\tfrac{n+q}{n-q}}}{V\left({L}_{1})}\\ & & \phantom{\rule[-1.5em]{}{0ex}}+\mu \frac{{\left[\frac{1}{n}{\displaystyle \int }_{{S}^{n-1}}{\left(\frac{{h}_{Q}}{{h}_{K}}\right)}^{p}\left({\alpha }_{K}\left(u)){\rho }_{K}^{q}\left(u){\rho }_{{L}_{2}}^{n-q}\left(u){\rm{d}}u\right]}^{\tfrac{n+q}{n-q}}}{V\left({L}_{2})}\\ & =& \lambda \frac{{\widetilde{V}}_{p,q}{\left(K,Q,{L}_{1})}^{\tfrac{n+q}{n-q}}}{V\left({L}_{1})}+\mu \frac{{\widetilde{V}}_{p,q}{\left(K,Q,{L}_{2})}^{\tfrac{n+q}{n-q}}}{V\left({L}_{2})}.\end{array}This yields inequality (3.5).By the equality condition of Minkowski’s integral inequality, we see that equality holds in (3.5) if and only if L1{L}_{1}and L2{L}_{2}are dilates.□Proof of Theorem 1.4Because of 0<q<n0\lt q\lt n, thus n+qn−q>0\frac{n+q}{n-q}\gt 0. Hence, by (1.6) and (3.5) we have ωnpnG˜p,q(K,λ∗L1+^qμ∗L2)n+qn−qV(λ∗L1+^qμ∗L2)=inf[nV˜p,q(K,Q,λ∗L1+^qμ∗L2)]n+qn−qV(λ∗L1+^qμ∗L2)V(Q∗)pnn+qn−q:Q∈Kon≥λinf[nV˜p,q(K,Q,L1)]n+qn−qV(L1)V(Q∗)pnn+qn−q:Q∈Kon+μinf[nV˜p,q(K,Q,L2)]n+qn−qV(L2)V(Q∗)pnn+qn−q:Q∈Kon=λωnpnG˜p,q(K,L1)n+qn−qV(L1)+μωnpnG˜p,q(K,L2)n+qn−qV(L2),\begin{array}{rcl}\frac{{\left[{\omega }_{n}^{\frac{p}{n}}{\widetilde{G}}_{p,q}\left(K,\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})\right]}^{\tfrac{n+q}{n-q}}}{V\left(\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}& =& \inf \left\{\frac{{\left[n{\widetilde{V}}_{p,q}\left(K,Q,\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})]}^{\tfrac{n+q}{n-q}}}{V\left(\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}{\left[V{\left({Q}^{\ast })}^{\frac{p}{n}}\right]}^{\tfrac{n+q}{n-q}}:Q\in {{\mathcal{K}}}_{o}^{n}\right\}\\ & \ge & \lambda \inf \left\{\frac{{\left[n{\widetilde{V}}_{p,q}\left(K,Q,{L}_{1})]}^{\tfrac{n+q}{n-q}}}{V\left({L}_{1})}{\left[V{\left({Q}^{\ast })}^{\frac{p}{n}}\right]}^{\tfrac{n+q}{n-q}}:Q\in {{\mathcal{K}}}_{o}^{n}\right\}\\ & & +\mu \inf \left\{\frac{{\left[n{\widetilde{V}}_{p,q}\left(K,Q,{L}_{2})]}^{\tfrac{n+q}{n-q}}}{V\left({L}_{2})}{\left[V{\left({Q}^{\ast })}^{\frac{p}{n}}\right]}^{\tfrac{n+q}{n-q}}:Q\in {{\mathcal{K}}}_{o}^{n}\right\}\\ & =& \lambda \frac{{\left[{\omega }_{n}^{\frac{p}{n}}{\widetilde{G}}_{p,q}\left(K,{L}_{1})\right]}^{\tfrac{n+q}{n-q}}}{V\left({L}_{1})}+\mu \frac{{\left[{\omega }_{n}^{\frac{p}{n}}{\widetilde{G}}_{p,q}\left(K,{L}_{2})\right]}^{\tfrac{n+q}{n-q}}}{V\left({L}_{2})},\end{array}i.e., G˜p,q(K,λ∗L1+^qμ∗L2)n+qn−qV(λ∗L1+^qμ∗L2)≥λG˜p,q(K,L1)n+qn−qV(L1)+μG˜p,q(K,L2)n+qn−qV(L2).\frac{{\widetilde{G}}_{p,q}{\left(K,\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}^{\tfrac{n+q}{n-q}}}{V\left(\lambda \ast {L}_{1}{\widehat{+}}_{q}\mu \ast {L}_{2})}\ge \lambda \frac{{\widetilde{G}}_{p,q}{\left(K,{L}_{1})}^{\tfrac{n+q}{n-q}}}{V\left({L}_{1})}+\mu \frac{{\widetilde{G}}_{p,q}{\left(K,{L}_{2})}^{\tfrac{n+q}{n-q}}}{V\left({L}_{2})}.This gives inequality (1.10).According to the equality condition of inequality (3.5), we see that the equality holds in (1.10) if and only if L1{L}_{1}and L2{L}_{2}are dilates.□

Journal

Open Mathematics – de Gruyter

Published: Jan 1, 2022

Keywords: ( p , q )-mixed volume; ( p , q )-mixed geominimal surface area; monotonic inequality; isoperimetric inequality; Brunn-Minkowski-type inequality; 52A20; 52A39; 52A40

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