5-7维亲吻构型上的非凸单位边多胞形
Non-convex unit-edge polytopes on kissing configurations in dimensions 5-7
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中文总结 AI 辅助
本文研究5-7维中九个猜想最优非格点亲吻构型，证明它们均为单位边多胞形顶点集，其中八个非凸，通过分裂格点接触多胞形面构造，并定义折痕多胞形类别，所有陈述经精确算术验证。
中文摘要 AI 辅助
在维度5、6和7中，所有九个已知的猜想最优非格点亲吻构型都是仅具有单位边的多胞形的顶点集。这九个中的八个是非凸的，而接触多胞形（即相同点的凸包）具有更长的边。单位边多胞形的边恰好是构型的接触。九个中除两个外，其余均由相同维度的格点接触多胞形通过分裂其某些面并重新组装碎片而构造。这是一种几何构造，不同于按层构造。分裂时折叠的面是Gosset系列$k_{21}$的连续成员，维度$n$中的分裂折叠为内积$1/(10-n)$：$1/5$、$1/4$或$1/3$。这些是接触多胞形与格点多胞形不同的内积。所有12个单位边多胞形（九个和三个格点的）都是折痕的，我们定义的这个类别通过允许不超过一半的浅折叠来扩展凸性。每个都是其顶点集上唯一的折痕单位边多胞形。每个陈述都在精确算术中得到验证。
英文摘要
All nine known conjecturally optimal non-lattice kissing configurations in dimensions 5, 6, and 7 are the vertex sets of polytopes with only unit edges. Eight of these polytopes are non-convex, and the contact polytopes, the convex hulls of the same points, have longer edges. The edges of the unit-edge polytopes are exactly the contacts of the configuration. All but two of the nine are constructed from the lattice contact polytope in the same dimension by splitting some of its facets and reassembling the pieces. This describes the configurations by the facets of a polytope rather than by layers. The facets that fold when split are consecutive members of the Gosset series $k_{21}$, and a split in dimension $n$ folds to the inner product $1/(10-n)$: $1/5$, $1/4$, or $1/3$. These are the inner products by which the contact polytopes differ from the lattice one. In dimensions 5 and 6 each unit-edge polytope is the only one on its vertex set. In dimension 7 uniqueness is proved within a class we define, the creased polytopes, which extends convexity by letting a facet lie on a hyperplane that cuts through the configuration, provided the facet contains every vertex on that hyperplane on its side of one of its ridges. Every statement is certified in exact arithmetic.
