ON LP-ORIENTATIONS OF CUBES AND CROSSPOLYTOPES
ON LP-ORIENTATIONS OF CUBES AND CROSSPOLYTOPES
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关于立方体和交叉多面体的 LP 方向
DOI：
10.1515/advg.2004.4.4.459
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发表时间：
2002
影响因子：
0.5
通讯作者：
M. Develin
中科院分区：
文献类型：
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作者：
M. Develin
在1996年的一次会议上发表的一篇论文中，Holt和Klee引入了一组必要条件，即d-多面体的图的定向是通过实现到Rn和该空间上的线性泛函而诱导的。一般说来，对于多面体P，是否其图中满足这些条件的每个方向都能以这种方式实现，这是一个悬而未决的问题；要考虑的两个自然多面体家族是立方体和交叉多面体。对于立方体，我们证明了，随着n的增长，可以实现的n-立方体Holt-Klee取向的百分比渐近为0。对于交叉抛物，我们给出了一组较强的条件，这些条件既是有向的充要条件，又是有向的充要条件；作为推论，我们证明了立方体的所有壳都是线壳。
In a paper presented at a 1996 conference, Holt and Klee introduced a set of necessary conditions for an orientation of the graph of a d-polytope to be induced by a realization into R n and linear functional on that space. In general, it is an open question to decide whether for a polytope P every orientation of its graph satisfying these conditions can in fact be realized in this fashion; two natural families of polytopes to consider are cubes and crosspolytopes. For cubes, we show that, as n grows, the percentage of n-cube Holt-Klee orientations which can be realized goes asymptotically to 0. For crosspolytopes, we give a stronger set of conditions which are both necessary and sufficient for an orientation to be in this class; as a corollary, we prove that all shellings of cubes are line shellings.