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arXiv · 0909.0471

A cubical antipodal theorem

Kyle E. Kinneberg·Aaron Mazel-Gee·Tia Sondjaja·Francis Edward Su
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Abstract

The classical Lusternik-Schnirelman-Borsuk theorem states that if a d-sphere is covered by d+1 closed sets, then at least one of the sets must contain a pair of antipodal points. In this paper, we prove a combinatorial version of this theorem for hypercubes. It is not hard to show that for any cover of the facets of a d-cube by d sets of facets, at least one such set contains a pair of antipodal ridges. However, we show that for any cover of the ridges of a d-cube by d sets of ridges, at least one set must contain a pair of antipodal k-faces, and we determine the maximum k for which this must occur, for all dimensions except d=5.

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Kyle E. Kinneberg, Aaron Mazel-Gee, Tia Sondjaja, Francis Edward Su. 2009-09-02. A cubical antipodal theorem. https://arxiv.org/abs/0909.0471

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