A Sparse Colorful Polytopal KKM Theorem

Abstract

Recently Soberón (Proc. Am. Math. Soc. Ser. B 9, 404–414 (2022)) proved a far-reaching generalization of the colorful KKM Theorem due to Gale (Int. J. Game Theory 13(1), 61–64 (1984)): let $n\ge k$, and assume that a family of closed sets $(A^i_j\,|\,i\in [n],\,j\in [k])$ has the property that for every $I\in \left( {\begin{array}{c}[n]\\ n-k+1\end{array}}\right)$, the family $\bigl (\bigcup _{i\in I}A^i_1,\dots ,\bigcup _{i\in I}A^i_k\bigr )$ is a KKM cover of the $(k-1)$-dimensional simplex $\Delta ^{k-1}$; then there is an injection $\pi :[k] \rightarrow [n]$ such that $\bigcap _{j=1}^k A_j^{\pi (j)}\ne \emptyset$. We prove a polytopal generalization of this result, answering a question of Soberón 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 Carathéodory theorem.