Positivity 13 (2009), 717–733
2009 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040717-17, published online February 20, 2009
The modiﬁed complex Busemann-Petty
problem on sections of convex bodies
Abstract. The complex Busemann-Petty problem asks whether origin
symmetric convex bodies in C
with smaller central hyperplane sections nec-
essarily have smaller volume. The answer is aﬃrmative if n ≤ 3 and negative
if n ≥ 4. Since the answer is negative in most dimensions, it is natural to
ask what conditions on the (n − 1)-dimensional volumes of the central sec-
tions of complex convex bodies with complex hyperplanes allow to compare
the n-dimensional volumes. In this article we give necessary conditions on the
section function in order to obtain an aﬃrmative answer in all dimensions.
The result is the complex analogue of .
Mathematics Subject Classiﬁcation (2000). 52A20.
Keywords. Convex bodies, sections, Fourier transform.
The Busemann-Petty problem was completely solved in the late 90’s as a result of a
series of papers of many mathematicians ([17,1,9,2,18,19], [4–7], [21,10,11,22, 7];
see [14, p.3] for the history of the solution). The problem asks the following:
Suppose K and L are two origin symmetric convex bodies in R
for every ξ ∈ S
K ∩ ξ
L ∩ ξ
Does it follow that
(K) ≤ Vol
The problem has an aﬃrmative answer only if n ≤ 4. Since the answer is negative
in most dimensions, it is natural to ask what conditions on the (n−1)-dimensional
volumes of central sections do allow to compare the n-dimensional volumes. Such
conditions were found in . The result is as follows.
For an origin symmetric convex body K in R
deﬁne the section function
(K ∩ ξ
Suppose K and L are origin symmetric convex smooth bodies in R
and α ∈ R
with α ≥ n − 4. Then, the inequality
(ξ) ≤ (−∆)