# Busemann's intersection inequality in hyperbolic and spherical spaces

## TL;DR

This paper extends Busemann's intersection inequality to hyperbolic and spherical geometries, identifying maximizers among convex bodies and exploring the inequality's behavior in different measure spaces.

## Contribution

It generalizes Busemann's intersection inequality to non-Euclidean spaces and analyzes the maximizers in hyperbolic and spherical geometries.

## Key findings

- Centered ellipsoids maximize the integral in hyperbolic and spherical spaces.
- The inequality's properties differ from the Euclidean case in curved geometries.
- The study includes analysis in general measure spaces.

## Abstract

Busemann's intersection inequality asserts that the only maximizers of the integral $\int_{S^{n-1}} |K\cap\xi^\perp|^n d\xi$ among all convex bodies of a fixed volume in $\mathbb R^n$ are centered ellipsoids. We study this question in the hyperbolic and spherical spaces, as well as general measure spaces.