A short proof of Gr\"unbaum's Conjecture about affine invariant points

TL;DR

This paper presents a simpler and shorter proof of Gr"unbaum's conjecture, which states that the set of affine invariant points of a convex body equals the set of points fixed by all affine symmetries of that body.

Contribution

The paper provides a new, more straightforward proof of Gr"unbaum's conjecture using topological methods, improving upon previous proofs.

Findings

Confirmed the equality of affine invariant points and fixed points for convex bodies.

Simplified the proof of Gr"unbaum's conjecture using topology.

Enhanced understanding of the structure of 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 $\mathcal K_n$.