The Lower Dimensional Busemann-Petty Problem in the Complex Hyperbolic Space

TL;DR

This paper investigates the lower dimensional Busemann-Petty problem in complex hyperbolic space, establishing that the volume comparison holds only for complex dimension one sections and not for higher dimensions.

Contribution

It extends the analysis of the Busemann-Petty problem to complex hyperbolic space and determines the dimensions for which the volume comparison is valid.

Findings

Affirmative answer for complex dimension one sections.

Negative answer for sections of higher complex dimensions.

Clarifies the open cases for k=2,3 in complex hyperbolic space.

Abstract

The lower dimensional Busemann-Petty problem asks whether origin-symmetric convex bodies in R^n with smaller volume of all k-dimensional sections necessarily have smaller volume. The answer is negative for k>3. The problem is still open for k=2,3. We study this problem in the complex hyperbolic n-space and prove that the answer is affirmative only for sections of complex dimension one and negative for sections of higher dimensions.