5维棱柱体的宽度
The width of 5-dimensional prismatoids
摘要：
桑托斯对赫希猜想（2012）的反例的构建是基于存在宽度大于d的维数d的棱柱体。Santos、Stephen和Thomas（2012）已经证明，这在$d \le 4$中不会发生。出于这一动机，我们在这里研究的宽度5维prismatoids，获得以下结果：-有5-prismatoids的宽度6只有25个顶点，而48个顶点在桑托斯的原始建设。这导致维度20的非赫希多面体，而不是原始维度43。- 对于任意大的$n$，存在具有$n$顶点和宽度$Ω(\sqrt{n})$的5-棱柱体。因此，5-prismatoids的宽度是无限的。
英文摘要：
Santos' construction of counter-examples to the Hirsch Conjecture (2012) is based on the existence of prismatoids of dimension d of width greater than d. Santos, Stephen and Thomas (2012) have shown that this cannot occur in $d \le 4$. Motivated by this we here study the width of 5-dimensional prismatoids, obtaining the following results: - There are 5-prismatoids of width six with only 25 vertices, versus the 48 vertices in Santos' original construction. This leads to non-Hirsch polytopes of dimension 20, rather than the original dimension 43. - There are 5-prismatoids with $n$ vertices and width $Ω(\sqrt{n})$ for arbitrarily large $n$. Hence, the width of 5-prismatoids is unbounded.
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