一个三次对映定理
A cubical antipodal theorem
摘要：
经典的Lusternik-Schnirelman-Borsuk定理指出，如果一个d-球面被d+1个闭集覆盖，那么至少有一个闭集必须包含一对对径点。在本文中，我们证明了这个定理的一个组合版本的超立方体。不难证明，对于d-立方体的任何小平面被d组小平面覆盖，至少有一组这样的小平面包含一对映脊。然而，我们表明，对于任何覆盖的脊的d-立方体的d组脊，至少有一组必须包含一对对映k-面，我们确定的最大k，这必须发生，除了d=5的所有尺寸。
英文摘要：
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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