The modified complex Busemann-Petty problem on sections of convex bodies

TL;DR

This paper investigates conditions on section volumes of complex convex bodies that could lead to positive volume comparison results across all dimensions, addressing a modified version of the complex Busemann-Petty problem.

Contribution

It provides necessary conditions on the section function to ensure affirmative volume comparison in the complex Busemann-Petty problem across all dimensions.

Findings

Identifies necessary conditions for the section function.

Addresses the modified complex Busemann-Petty problem.

Provides insights into volume comparison criteria.

Abstract

Since the answer to the complex Busemann-Petty problem 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.