The affine Tverberg theorem revisited

Abstract

We prove the Bárány--Kalai--Tverberg conjecture extending the affine version of Tverberg's theorem to simplicial balls and polytopes other than the simplex: For any affine map $\varphi :P\to {\mathbb{R}}^d$ from a convex polytope $P$ of dimension $N=(d+1)(r-1)$ with $r\ge 2$, there exist $r$ pairwise disjoint faces $F_1,\dots ,F_r\subset \partial P$ such that $\varphi (F_1)\cap \cdots \cap \varphi (F_r)\ne \varnothing$. A similar statement holds for a simplicial ball $P$ with $\varphi$ assumed affine on all of its faces.