A Sparse colorful polytopal KKM Theorem

Abstract: Recently Sober\'on proved a far-reaching generalization of the colorful KKM Theorem due to Gale: let $n\geq k$, and assume that a family of closed sets $(A^i_j\mid i\in [n], j\in [k])$ has the property that for every $I\in \binom{[n]}{n-k+1}$, the family $$\big(\bigcup_{i\in I}A^{i_1,\dots,\bigcup_{i\in} I}A<sup>i_k\big)$$ is a KKM cover of the $(k-1)$-dimensional simplex $\Delta<sup>{k-1}$; then there is an injection $\pi:[k] \rightarrow [n]$ so that $$\bigcap_{i=1}^k A_i^{\pi(i)}\neq \emptyset$$. We prove a polytopal generalization of this result, answering a question of Sober\'on in the same note. We also discuss applications of our theorem to fair division of multiple cakes, $d$-interval piercing, and a generalization of the colorful Carath\'eodory theorem.