The topological KKMS Theorem is a powerful extension of the Brouwer’s Fixed-Point Theorem, which was proved by Shapley in 1973 in the context of game theory.<\/p>\n
We prove a colorful and polytopal generalization of the KKMS Theorem, and show that our theorem implies some seemingly unrelated results in discrete geometry and combinatorics involving colorful settings.<\/p>\n
For example, we apply our theorem to provide a new proof of the Colorful Caratheodory Theorem due to Barany, which asserts that if 0 is in the convex hull of n+1 sets of points in R^n, then there exists a colorful selection of points, one from each set, containing 0 in its convex hull. We further apply our theorem to obtain an upper bound on the piercing numbers in colorful d-interval families, extending results of Tardos, Kaiser and Alon for the non-colored case. Finally, we apply our theorem to questions regarding envy-free fair division of goods (e.g., cakes) among a set of players.<\/p>\n
Joint with Florian Frick.<\/p>\n","protected":false},"excerpt":{"rendered":"
The topological KKMS Theorem is a powerful extension of the Brouwer’s Fixed-Point Theorem, which was proved by Shapley in 1973 in the context of game theory. We prove a colorful and polytopal generalization of the KKMS Theorem, and show that…
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