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

Let us denote by K n the hyperspace of all convex bodies of R n equipped with the Hausdorff distance topology. An affine invariant point p is a continuous and Aff ( n ) -equivariant map p : K n → R n , where Aff ( n ) denotes the group of all nonsingular affine maps of R n . For every K ∈ K n , let P n ( K ) = { p ( K ) ∈ R n | p is an affine invariant point } and F n ( K ) = { x ∈ R n | g x = x for every g ∈ Aff ( n ) such that g K = K } . In 1963, B. Grünbaum conjectured that P n ( K ) = F n ( 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 ) on K n . [ABSTRACT FROM AUTHOR]