A cubical antipodal theorem
Combinatorics
2009-09-03 v1 Geometric Topology
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.
Cite
@article{arxiv.0909.0471,
title = {A cubical antipodal theorem},
author = {Kyle E. Kinneberg and Aaron Mazel-Gee and Tia Sondjaja and Francis Edward Su},
journal= {arXiv preprint arXiv:0909.0471},
year = {2009}
}
Comments
15 pages, 2 figures. See related work at http://www.math.hmc.edu/~su/papers.html