将正八面体内接于多面体

旗单纯3-多面体的组合学

Combinatorics of flag simplicial 3-polytopes

本文的重点是边收缩操作。我们可以证明单纯3-多面体是旗当且仅当它的任何边的收缩给出单纯3-多面体。我们的主要结果表明，任何旗形单纯3-多面体都可以通过边收缩序列化为八面体。利用这一运算，我们在旗单纯3-多面体集上引入了一个偏序，并研究了相应偏序集上的Hasse图。我们估计输入和输出度的顶点的Hasse图。

Facet volumes of polytopes

In this paper, motivated by the work of Edelman and Strang, we show that for fixed integers $d\geq 2$ and $n\geq d+1$ the configuration space of all facet volume vectors of all $d$-polytopes in $\mathbb R^{d}$ with $n$ facets is a full dimensional cone in $\mathbb R^{n}$. In particular, for tetrahedra ($d=3$ and $n=4$) this is a cone over a regular octahedron. Our proof is based on a novel configuration space / test map scheme which uses topological methods for finding solutions of a problem, and tools of differential geometry to identify solutions with the desired properties. Furthermore, our…