许多等射影多面体
Many equiprojective polytopes
AI总结：
本文针对等射影多面体的分类问题，利用Goodman--Pollack序型下界与Minkowski和构造新方法，证明k趋于无穷时k-等射影多面体的组合类型数至少为k的3k/2+o(k)次方，大幅改进了此前的线性下界结果。
摘要：
当三维多面体$P$沿任意不平行于其面的直线的投影都是一个具有$k$个顶点的多边形时，称$P$是$k$-等射影的。1968年，Geoffrey Shephard提出了描述所有等射影多面体的问题。近期研究表明，$k$-等射影多面体的组合类型数量至少随$k$呈线性增长。本文证明，当$k$趋于无穷时，这类组合类型的数量至少为$k^{3k/2+o(k)}$。该结果依赖于Goodman--Pollack关于序型数量的下界，以及通过Minkowski和构造等射影多面体的新方法。
英文摘要：
A $3$-dimensional polytope $P$ is $k$-equiprojective when the projection of $P$ along any line that is not parallel to a facet of $P$ is a polygon with $k$ vertices. In 1968, Geoffrey Shephard asked for a description of all equiprojective polytopes. It has been shown recently that the number of combinatorial types of $k$-equiprojective polytopes is at least linear as a function of $k$. Here, it is shown that there are at least $k^{3k/2+o(k)}$ such combinatorial types as $k$ goes to infinity. This relies on the Goodman--Pollack lower bound on the number of order types and on new constructions of equiprojective polytopes via Minkowski sums.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
