Colorful coverings of polytopes and piercing numbers of colorful d-intervals

We prove a common strengthening of B\'ar\'any's colorful Carath\'eodory theorem and the KKMS theorem. In fact, our main result is a colorful polytopal KKMS theorem, which extends a colorful KKMS theorem due to Shih and Lee [Math. Ann. 296 (1993), no. 1, 35--61] as well as a polytopal KKMS theorem due to Komiya [Econ. Theory 4 (1994), no. 3, 463--466]. The (seemingly unrelated) colorful Carath\'eodory theorem is a special case as well. We apply our theorem to establish an upper bound on the piercing number of colorful d-interval hypergraphs, extending earlier results of Tardos [Combinatorica 15 (1995), no. 1, 123--134] and Kaiser [Discrete Comput. Geom. 18 (1997), no. 2, 195--203].