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- W200980779 abstract "In this short note, we show that if P is a d-polytope which is not the simplex, then for all 0 2. Since the statement is symmetric in k and d − k, we can assume k 1. Take some vertex figure P/v which is not a simplex. By induction, we can find disjoint faces in P/v of dimensions k and d−k−1. These correspond to faces G and H of P , of respective dimension k + 1 and d− k, which intersect only in v. However, taking any facet of G not containing v, we obtain a face of dimension k which is disjoint from H, completing the proof. Given Theorem 1, the natural question to ask is the following. Question 1. What can we say about k-faces which intersect all (d−k)-faces of a d-polytope, or about polytopes which have such faces? For k = d − 1, for instance, the polytope P must be a pyramid over the face in question. However, even for the case k = 1 and d = 3, there are many polytopes which have such a k-face: for instance, we can start with the simplex and a distinguished edge, and repeatedly add points beyond a 2-face containing the edge in question. Acknowledgements. I would like to thank Ezra Miller and Vic Reiner for bringing the problem to my attention, as well as the American Instititute of Mathematics and the University of Minnesota for their financial support." @default.
- W200980779 created "2016-06-24" @default.
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- W200980779 date "2004-01-01" @default.
- W200980779 modified "2023-09-23" @default.
- W200980779 title "Disjoint Faces of Complementary Dimension" @default.
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