On a problem concerning affine-invariant points of convex sets

TL;DR

This paper discusses affine-invariant points of convex sets, providing solutions to old problems by extending the group considerations to compact cases, and confirms a conjecture related to similarity-invariant points.

Contribution

It offers a complete solution to B. Grünbaum's problem by restricting to compact groups and proves the existence of invariant points under these conditions.

Findings

Existence of invariant points for convex bodies under compact group actions

Proof of Grünbaum's conjecture for similarity-invariant points

An example showing the set of all such points cannot be finitely generated

Abstract

This text is a somewhat reformatted (e.g., some statements that were not as such in the original paper, are given the names "Corollary" or "Theorem.") translation of the old and practically inaccessible paper: P. Kuchment, On the question of the affine-invariant points of convex bodies, (in Russian), Optimizacija No. 8(25) (1972), 48--51, 127. MR0350621. There partial solutions of some old problems of B. Gr\"unbaum concerning affine-invariant points of convex bodies were obtained. The main restriction, due to which the solution was incomplete, was the compactness restriction on the group of linear transformations involved. It was noticed recently by O. Mordhorst (arXiv:1601.07850) that a simple additional argument allows one to restrict the consideration to the bodies whose John's ellipsoid is a ball and consequently to a compact group case. This in turn extends the result to the complete solution of B. Gr\"unbaum's problem (see also some partial progress in the previous recent works by M. Meyer, C. Sch\"utt, and E. Werner). The article translated below contains three main statements: existence of invariant points in the case when the linear parts of affine transformations are taken from a compact group, the resulting from this proof of B. Gr\"unbaum's conjecture for the case of similarity invariant points, and an example precluding the set of all such points from being the convex hull of finitely many of them.