  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.
