The Busemann-Petty Problem in Complex Hyperbolic Space

TL;DR

This paper investigates the Busemann-Petty problem within complex hyperbolic space, establishing the dimensional thresholds where the answer shifts from affirmative to negative, extending known Euclidean results to complex hyperbolic geometry.

Contribution

It extends the Busemann-Petty problem to complex hyperbolic space, determining the dimensions where the problem's answer is affirmative or negative.

Findings

Affirmative for n ≤ 2 in complex hyperbolic space

Negative for n ≥ 3 in complex hyperbolic space

Extends Euclidean results to complex hyperbolic geometry

Abstract

The Busemann-Petty problem asks whether origin-symmetric convex bodies in real Euclidean n-space with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative for n less or equal to 4 and negative if n greater or equal to 5. We study this problem in the complex hyperbolic n-space and prove that the answer is affirmative for n less or equal to 2 and negative for n greater or equal to 3.