论文标题
紧凑型双曲线$ d $ d $ d $ d+4 $ facets和相关尺寸范围的近乎分类
Near Classification of Compact Hyperbolic Coxeter $d$-Polytopes with $d+4$ Facets and Related Dimension Bounds
论文作者
论文摘要
我们完成了紧凑型双曲线coxeter $ d $ - polytopes的分类，其中$ d+4 $ facets in $ d = 4 $和$ 5 $。根据Felikson和Tumarkin的先前工作，可能会出现新的多型的唯一剩余维度是$ d = 6 $。我们得出了一种通过点设置顺序类型的分类来生成这些多层的组合类型的新方法。在尺寸$ 4 $和$ 5 $中，分别有$ 348 $和$ 51 $ polytopes，为进一步研究提供了许多新示例。 我们此外，在$ d+k $ facets $ k \ leq 10 $的尺寸$ d $上提供新的上限。文伯格（Vinberg）在1985年表明，对于任何$ k $，我们都有$ d \ leq 29 $，并且以前没有以$ k \ geq 5 $发布更好的界限。由于我们的界限，我们证明了紧凑的双曲线coxeter $ 29 $ -Polytope至少具有$ 40 $的尺寸。
We complete the classification of compact hyperbolic Coxeter $d$-polytopes with $d+4$ facets for $d=4$ and $5$. By previous work of Felikson and Tumarkin, the only remaining dimension where new polytopes may arise is $d=6$. We derive a new method for generating the combinatorial type of these polytopes via the classification of point set order types. In dimensions $4$ and $5$, there are $348$ and $51$ polytopes, respectively, yielding many new examples for further study. We furthermore provide new upper bounds on the dimension $d$ of compact hyperbolic Coxeter polytopes with $d+k$ facets for $k \leq 10$. It was shown by Vinberg in 1985 that for any $k$, we have $d \leq 29$, and no better bounds have previously been published for $k \geq 5$. As a consequence of our bounds, we prove that a compact hyperbolic Coxeter $29$-polytope has at least $40$ facets.
