New results on affine invariant points

TL;DR

This paper proves a conjecture relating affine invariant points to affine symmetries of convex bodies, explores the structure of affine invariant points, and examines extremal symmetry cases.

Contribution

It confirms Grünbaum's conjecture, demonstrates the infinite dimensionality of affine linear points, and analyzes extremal symmetry measures in convex bodies.

Findings

Affine invariant points coincide with points invariant under all affine symmetries.

The affine space of affine linear points is infinite dimensional.

The distance between centers of John and Löwner ellipsoids can be arbitrarily large.

Abstract

We prove a conjecture of B. Gr\"unbaum stating that the set of affine invariant points of a convex body equals to the set of points invariant under all affine linear symmetries of the convex body. As a consequence we give a short proof on the fact that the affine space of affine linear points is infinite dimensional. In particular, we show that the set of affine invariant points with no dual is of second category. We investigate extremal cases for a class of symmetry measures. We show that the center of the John and L\"owner ellipsoid can be far apart and we give the optimal order for the extremal distance of the two centers.