The complex Busemann-Petty problem on sections of convex bodies

TL;DR

This paper investigates the complex Busemann-Petty problem, determining that for dimensions up to 3 the answer is yes, and for dimensions 4 and above, the answer is no, based on convex body sections in complex space.

Contribution

The paper establishes the dimension-dependent affirmative and negative answers to the complex Busemann-Petty problem for convex bodies.

Findings

Affirmative answer for n ≤ 3

Negative answer for n ≥ 4

Dimension-dependent behavior of the problem

Abstract

The complex Busemann-Petty problem asks whether origin symmetric convex bodies in $\C^n$ with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if $n\le 3$ and negative if $n\ge 4.$