Improved Tverberg theorems for certain families of polytopes

A theorem of Gr\"unbaum, which states that every $m$-polytope is a refinement of an $m$-simplex, implies the following generalization of Tverberg's theorem: if $f$ is a linear function from an $m$-dimensional polytope $P$ to $\mathbb{R}^d$ and $m \ge (d + 1)(r - 1)$, then there are $r$ pairwise disjoint faces of $P$ whose images intersect. Moreover, the topological Tverberg theorem implies that this statement is true whenever the map $f$ is continuous and $r$ is a prime power. In this note, we show that for certain families of polytopes the lower bound on the dimension $m$ of the polytopes can be significantly improved, both in the affine and topological cases.