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arXiv · 1412.8621

Colorful versions of the Lebesgue, KKM, and Hex theorem

Djordje Baralić·Rade Živaljević
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Abstract

Following and developing ideas of R. Karasev (Covering dimension using toric varieties, arXiv:1307.3437), we extend the Lebesgue theorem (on covers of cubes) and the Knaster-Kuratowski-Mazurkiewicz theorem (on covers of simplices) to different classes of convex polytopes (colored in the sense of M. Joswig). We also show that the $n$-dimensional Hex theorem admits a generalization where the $n$-dimensional cube is replaced by a $n$-colorable simple polytope. The use of quasitoric manifolds offers great flexibility and versatility in applying the general method.

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Djordje Baralić, Rade Živaljević. 2015-02-12. Colorful versions of the Lebesgue, KKM, and Hex theorem. https://arxiv.org/abs/1412.8621

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