Neighborly Cubical Polytopes
Neighborly Cubical Polytopes
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邻方多面体
DOI：
10.1007/s004540010039
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发表时间：
1998
影响因子：
0.8
通讯作者：
G. Ziegler
中科院分区：
文献类型：
--
作者：
M. Joswig;G. Ziegler
摘要。邻立方多面体存在：对于任意n≥d≥2r+2，存在一个立方凸d多面体Cdn，其r骨架与n维立方体的r骨架组合等效。这就解决了巴布森、比莱拉和陈的问题。卡莱推测，边界
邻近立方多面体Cdn的$\partial C_d^n$在所有有2n个顶点的立方（d-1）球中最大化f向量。虽然我们证明了如果n≤d+1，这对于多面球是正确的，但我们也给出了d=4和n=6的反例。此外，邻立方多面体的存在表明，n维立方体的图（其中n \ge 5）在gr<s:1> nbaum意义上是“维度模糊的”。我们还证明了5立方的图是“强模糊”的。在d=4的特殊情况下，相邻的立方体多面体有f3=(f0/4) log2 （f0/4）个顶点，因此面顶点比f3/f0是无界的；这就解决了Jockusch研究过的Kalai、Perles和Stanley的一个问题。
Abstract. Neighborly cubical polytopes exist: for any n≥ d≥ 2r+2 , there is a cubical convex d -polytope Cdn whose r -skeleton is combinatorially equivalent to that of the n -dimensional cube. This solves a problem of Babson, Billera, and Chan. Kalai conjectured that the boundary
$\partial C_d^n$ of a neighborly cubical polytope Cdn maximizes the f -vector among all cubical (d-1) -spheres with 2n vertices. While we show that this is true for polytopal spheres if n≤ d+1 , we also give a counterexample for d=4 and n=6 . Further, the existence of neighborly cubical polytopes shows that the graph of the n -dimensional cube, where n\ge5 , is ``dimensionally ambiguous'' in the sense of Grünbaum. We also show that the graph of the 5 -cube is ``strongly 4 -ambiguous.'' In the special case d=4 , neighborly cubical polytopes have f3=(f0/4) log2 (f0/4) vertices, so the facet—vertex ratio f3/f0 is not bounded; this solves a problem of Kalai, Perles, and Stanley studied by Jockusch.