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 n with smaller central hyperplane sections necessarily 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 sections 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 [16].

Comparison of volumes of convex bodies in real, complex, and quaternionic spaces

“…Smoothness properties of these functions play a decisive role in establishing main results, and we study them in detail. Similar properties in the context of the modified BP problem in R n and C n were briefly indicated in [36,35,74], however, the details (which are important and fairly nontrivial) were omitted.…”

“…An important feature of these operators is that the corresponding Fourier multiplier |y| α does not preserve the Schwartz space S(R N ) and the phrases like "in the sense of distributions" (cf. [36,35,74]) require careful explanation and justification. Section 4 is devoted to weighted section functions of origin-symmetric convex bodies.…”

“…It was called the modified Busemann-Petty problem. Both questions were studied by Koldobsky, König and Zymonopoulou [34] and Zymonopoulou [74] for convex bodies in C n . The answer to the first question for C n is "Yes" if and only if n 3.…”

A quasi-Funk’s section theorem in n-complex space

“…[8], an extension was achieved as follows: If K is an origin-symmetric star body, L is a star body and for some i such that 0 , i n ∧ ∧ vol ( ) vol ( ) [9] and [10] introduced the complex intersections of star bodies in the complex n -space n C . Koldobsky's solution to complex version of Busemann-Petty problem [10][11][12][13][14][15][16] can be expressed by:…”

Volume inequalities for complex isotropic measures

“…The idea to find analogs of known results from Euclidean geometry in complex vector spaces is not new. In recent years, the study of convex bodies in C n has received considerable attention (see, e.g., [1][2][3][4][5]13,15,19,[22][23][24][25][26]33,34,37,41,42]). …”