A Sparse colorful polytopal KKM Theorem

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 $\big(\bigcup_{i\in I}A^i_1,\dots,\bigcup_{i\in I}A^i_k\big)$ is a KKM cover of the $(k-1)$-dimensional simplex $\Delta^{k-1}$; 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.