On the local convexity of intersection bodies of revolution

TL;DR

This paper investigates how the intersection body operation affects convexity of symmetric bodies of revolution, showing that in high dimensions, the double intersection body closely resembles an ellipsoid, indicating improved convexity.

Contribution

It provides new quantitative insights into the convexity properties of intersection bodies of revolution, especially in high dimensions, extending Busemann's theorem.

Findings

Intersection bodies of symmetric convex bodies of revolution behave like Euclidean balls near the equator.

In high dimensions, the double intersection body is close to an ellipsoid in Banach-Mazur distance.

The paper establishes local convexity properties of intersection bodies at the equator for star bodies of revolution.

Abstract

One of the fundamental results in Convex Geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior near the equator as the Euclidean ball. We apply this result to show that in sufficiently high dimension the double intersection body of a symmetric convex body of revolution is very close to an ellipsoid in the Banach-Mazur distance. We also prove results on the local convexity at the equator of intersection bodies in the class of star bodies of revolution.