The geometry of p-convex intersection bodies

TL;DR

This paper extends Busemann's theorem to p-convex bodies, showing their intersection bodies are q-convex, and explores the geometric implications and measure space extensions of these convexity properties.

Contribution

It provides a p-convex version of Busemann's theorem, establishes the convexity of intersection bodies in this setting, and analyzes their geometric and measure-theoretic properties.

Findings

Intersection bodies of p-convex bodies are q-convex for certain q.

Constructed examples demonstrate the sharpness of the convexity results.

Intersection bodies can be significantly farther from Euclidean balls than the original bodies.

Abstract

Busemann's theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper we provide a version of Busemann's theorem for p-convex bodies. We show that the intersection body of a p-convex body is q-convex for certain q. Furthermore, we discuss the sharpness of the previous result by constructing an appropriate example. This example is also used to show that IK, the intersection body of K, can be much farther away from the Euclidean ball than K. Finally, we extend these theorems to some general measure spaces with log-concave and $s$-concave measures