A cubical antipodal theorem

TL;DR

This paper extends the classical antipodal theorem from spheres to hypercubes, establishing conditions under which antipodal faces must appear in covers of hypercube facets and ridges, with a focus on combinatorial and topological properties.

Contribution

It provides a combinatorial version of the antipodal theorem for hypercubes, identifying the maximum dimension of antipodal faces guaranteed in covers, except for the case d=5.

Findings

For covers of hypercube facets, at least one set contains antipodal points.

For covers of hypercube ridges, at least one set contains antipodal k-faces.

Determines the maximum k for which antipodal k-faces are guaranteed, except when d=5.

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.