Parkable convex sets and finite-dimensional Hilbert spaces

TL;DR

This paper introduces the concept of parkability for convex sets in Euclidean spaces, providing characterizations of ellipsoids and symmetric convex bodies, and connects these to Hilbert space properties and illumination boundary characterizations.

Contribution

It offers new characterizations of ellipsoids and symmetric convex bodies using parkability, answering longstanding questions and extending Hilbert space characterizations.

Findings

Characterizations of ellipsoids in Euclidean spaces.

Characterizations of centrally symmetric convex bodies.

Improved boundary illumination results for ellipsoids.

Abstract

A subset of a convex body $B$ containing the origin in a Euclidean space is {\it parkable in $B$} if it can be translated inside $B$ in such a manner that the translate the origin. We provide characterizations of ellipsoids and of centrally symmetric convex bodies in Euclidean spaces of dimension $\ge 3$ based on the notion of parkability, answering several questions posed by G. Bergman. The techniques used, which are based on characterizations of Hilbert spaces among finite-dimensional Banach spaces in terms of their lattices of subspaces and algebras of endomorphisms, also apply to improve a result of W. Blaschke characterizing ellipsoids in terms of boundaries of illumination.