Affine invariant points

TL;DR

This paper investigates the structure of affine invariant points, proving the existence of a rich set of such points and their density in convex bodies, introducing new affine invariant points in the process.

Contribution

It provides a positive answer to Gruenbaum's question about the size of the set of affine invariant points and introduces new affine invariant points not previously studied.

Findings

The set of convex bodies with all affine invariant points covering the space is dense.

Existence of a rich set of affine invariant points.

Introduction of new affine invariant points.

Abstract

We answer in the negative a question by Gruenbaum who asked if there exists a finite basis of affine invariant points. We give a positive answer to another question by Gruenbaum about the "size" of the set of all affine invariant points. Related, we show that the set of all convex bodies K, for which the set of affine invariant points is all of n-dimensional Euclidean space, is dense in the set of convex bodies. Crucial to establish these results, are new affine invariant points, not previously considered in the literature.