Colorful versions of the Lebesgue, KKM, and Hex theorem
Metric Geometry
2015-02-13 v2
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 -dimensional Hex theorem admits a generalization where the -dimensional cube is replaced by a -colorable simple polytope. The use of quasitoric manifolds offers great flexibility and versatility in applying the general method.
Keywords
Cite
@article{arxiv.1412.8621,
title = {Colorful versions of the Lebesgue, KKM, and Hex theorem},
author = {Djordje Baralić and Rade Živaljević},
journal= {arXiv preprint arXiv:1412.8621},
year = {2015}
}