Reflections of convex bodies and their sections

TL;DR

This paper investigates how convex bodies, especially ellipsoids, can be reflected through their sections and extends these reflections to the entire body, leading to new characterizations and insights into longstanding conjectures.

Contribution

It introduces a novel reflection-based characterization of ellipsoids and explores their properties, providing new tools for understanding convex geometry.

Findings

Ellipsoids can be characterized by their reflection properties.

Reflections of sections can extend to reflections of the entire convex body.

Results relate to a conjecture by K. Bezdek.

Abstract

The purpose of this paper is to study the reflections of a convex body. In particular, we are interested in orthogonal reflections of its sections that can be extended to reflections of the whole body. For this reason, we need to study the case of a non-spherical ellipsoid, where a surprising structure arises (Section 2). These results allow us to give a new characterization of ellipsoids in terms of their reflections and, on the other hand, to prove a result deeply related to a conjecture due to K. Bezdek.