The affine Tverberg theorem revisited

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 $\phi :P\to {\mathbb{R}}^d$ from a convex polytope $P$ of dimension $N=(d+1)(r-1)$ with $r\ge 2$, there exist $r$ pairwise disjoint faces $F_1,\dots ,F_r\subset \mathrm{\partial }P$ such that $\phi (F_1)\cap \cdots \cap \phi (F_r)\mathrm{\ne }\mathrm{\varnothing }$. A similar statement holds for a simplicial ball $P$ with $\phi$ assumed affine on all of its faces.