Convex bodies all whose sections (projections) are equal

TL;DR

This paper investigates whether convex bodies with all equal hyperplane sections or projections must necessarily be spheres, exploring the relationship between symmetry and geometric properties in convex geometry.

Contribution

It provides new insights into conditions under which convex bodies with equal sections or projections are necessarily spherical, linking topology and convex geometry.

Findings

Convex bodies with all equal hyperplane sections are characterized.

Convex bodies with all equal orthogonal projections are analyzed.

Results connect symmetry properties with geometric and topological conditions.

Abstract

The purpose of this paper is to answer the following question: If all hyperplane sections through the origin of a convex body are "equal", is the convex body "equal" to the ball? The meaning of the notion "equal" will change in the course of this paper. Similarly, we are interested in the following problem: If all orthogonal projections of a convex body onto hyperplanes are "equal", is the convex body "equal" to the ball? Topology and convex geometry are deeply interrelated in the solution and understanding of these problems.