Star bodies with completely symmetric sections

TL;DR

This paper proves that if all central sections of a star body are completely symmetric, then the body must be a Euclidean ball, extending to functions on the sphere and addressing open questions in convex geometry.

Contribution

It establishes that complete symmetry in all sections implies the star body is a ball, and generalizes to functions on the sphere with isotropic restrictions.

Findings

Star bodies with symmetric sections are necessarily balls.

Functions with isotropic restrictions on almost all equators are constant.

Answers to open questions in convex geometry.

Abstract

We say that a star body $K$ is completely symmetric if it has centroid at the origin and its symmetry group $G$ forces any ellipsoid whose symmetry group contains $G$, to be a ball. In this short note, we prove that if all central sections of a star body $L$ are completely symmetric, then $L$ has to be a ball. A special case of our result states that if all sections of $L$ are origin symmetric and 1-symmetric, then $L$ has to be a Euclidean ball. This answers a question from \cite{R2}. Our result is a consequence of a general theorem that we establish, stating that if the restrictions in almost all equators of a real function $f$ defined on the sphere, are isotropic functions, then $f$ is constant a.e. In the last section of this note, applications, improvements and related open problems are discussed and two additional open questions from \cite{R} and \cite{R2} are answered.}