许多等射多边形

Pub Date : 2024-07-23 DOI:10.1007/s00454-024-00681-7
Théophile Buffière, Lionel Pournin
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引用次数: 0

摘要

当一个三维多边形 P 沿着与 P 的一个面不平行的任何线的投影是一个有 k 个顶点的多边形时,这个多边形 P 是 k 等投影的。1968 年,杰弗里-谢泼德(Geoffrey Shephard)要求描述所有等投影多面体。最近的研究表明,k 等投影多边形的组合类型数量至少是 k 的线性函数。这里的研究表明,当 k 变为无穷大时,至少有 \(k^{3k/2+o(k)}\) 个这样的组合类型。这依赖于古德曼-波拉克(Goodman-Pollack)关于点配置阶类型数量的下限,以及通过闵科夫斯基和对等投影多面体的新构造。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Many Equiprojective Polytopes

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Many Equiprojective Polytopes

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 of point configurations and on new constructions of equiprojective polytopes via Minkowski sums.

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