A short proof of Gr\"unbaum'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. Gr\"unbaum 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 $\mathcal K_n$.