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

Let us denote by KnKn the hyperspace of all convex bodies of RnRn equipped with the Hausdorff distance topology. An affine invariant point p   is a continuous and Aff(n)Aff(n)-equivariant map p:Kn→Rnp:Kn→Rn, where Aff(n)Aff(n) denotes the group of all nonsingular affine maps of RnRn. For every K∈KnK∈Kn, let Pn(K)={p(K)∈Rn|p is an affine invariant point}Pn(K)={p(K)∈Rn|p is an affine invariant point} and Fn(K)={x∈Rn|gx=x for every g∈Aff(n) such that gK=K}Fn(K)={x∈Rn|gx=x for every g∈Aff(n) such that gK=K}. In 1963, B. Grünbaum conjectured that Pn(K)=Fn(K)Pn(K)=Fn(K)[3]. After some partial results, the conjecture was recently proven in [6].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)Aff(n) on KnKn.