A short proof of Grünbaum's Conjecture about affine invariant points

Abstract

Let us denote by ${\mathcal{K}}_n$ the hyperspace of all convex bodies of ${\mathbb{R}}^n$ equipped with the Hausdorff distance topology. An affine invariant point $p$ is a continuous and Aff(n)-equivariant map $p:{\mathcal{K}}_n\to {\mathbb{R}}^n$, where Aff(n) denotes the group of all nonsingular affine maps of ${\mathbb{R}}^n$. For every $K\in {\mathcal{K}}_n$, let ${\mathfrak{P}}_n(K)=\{p(K)\in {\mathbb{R}}^n\mid p\text{ is an affine invariant point}\}$ and ${\mathfrak{F}}_n(K)=\{x\in {\mathbb{R}}^n\mid gx=x\text{ for every }g\in Aff(n)\text{ such that }gK=K\}$. In 1963, B. Grünbaum conjectured that ${\mathfrak{P}}_n(K)={\mathfrak{F}}_n(K)$ . After some partial results, the conjecture was recently proven by O. Mordhorst. In this short note we give a rather different, simpler and shorter proof of this conjecture, based merely on the topology of the action of Aff(n) on ${\mathcal{K}}_n$.