BIRS Workshop Lecture Videos
Colorful coverings of polytopes -- the hidden topological truth behind different colorful phenomena Zerbib, Shira
The topological KKMS Theorem is a powerful extension of 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 our theorem implies some seemingly unrelated results in discrete geometry and combinatorics involving colorful settings. For example, we apply our theorem to provide a new proof of the Colorful Caratheodory Theorem due to Barany, and also 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. We further apply our theorem to questions regarding envy-free fair division of goods among a set of players. Joint with Florian Frick.
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