{"title":"许多等射多边形","authors":"Théophile Buffière, Lionel Pournin","doi":"10.1007/s00454-024-00681-7","DOIUrl":null,"url":null,"abstract":"<p>A 3-dimensional polytope <i>P</i> is <i>k</i>-equiprojective when the projection of <i>P</i> along any line that is not parallel to a facet of <i>P</i> is a polygon with <i>k</i> 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 <i>k</i>-equiprojective polytopes is at least linear as a function of <i>k</i>. Here, it is shown that there are at least <span>\\(k^{3k/2+o(k)}\\)</span> such combinatorial types as <i>k</i> 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.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Many Equiprojective Polytopes\",\"authors\":\"Théophile Buffière, Lionel Pournin\",\"doi\":\"10.1007/s00454-024-00681-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A 3-dimensional polytope <i>P</i> is <i>k</i>-equiprojective when the projection of <i>P</i> along any line that is not parallel to a facet of <i>P</i> is a polygon with <i>k</i> 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 <i>k</i>-equiprojective polytopes is at least linear as a function of <i>k</i>. Here, it is shown that there are at least <span>\\\\(k^{3k/2+o(k)}\\\\)</span> such combinatorial types as <i>k</i> 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.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-07-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00454-024-00681-7\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00454-024-00681-7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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。
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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