Convex bodies all whose sections (projections) are equal

Abstract

This work deals with the following question: if all hyperplane sections through the origin (orthogonal projections) of a convex body are “equal”, is the convex body “equal” to the ball? where the notion of “equal” changes throughout the paper. Topology, Lie groups, Fourier analysis, and convex geometry interrelates in the solution and understanding of these problems.