On a generalization of Busemann's intersection inequality

TL;DR

This paper extends Busemann's intersection inequality to a broader class of star bodies, providing bounds for their intersection bodies near the Euclidean ball and improving existing bounds for certain dimensions.

Contribution

It generalizes Busemann's inequality to $k$-intersection bodies for star bodies close to the Euclidean ball and refines bounds for specific dimension ratios.

Findings

Established bounds for $k$-intersection bodies near the Euclidean ball.

Improved bounds for general star bodies when $k$ is proportional to the dimension.

Extended the applicability of Busemann's inequality to new classes of star bodies.

Abstract

Busemann's intersection inequality gives an upper bound for the volume of the intersection body of a star body in terms of the volume of the body itself. Koldobsky, Paouris, and Zymonopoulou asked if there is a similar result for $k$-intersection bodies. We solve this problem for star bodies that are close to the Euclidean ball in the Banach-Mazur distance. We also improve a bound obtained by Koldobsky, Paouris, and Zymonopoulou for general star bodies in the case when $k$ is proportional to the dimension.