具有指数照明数的恒宽凸面体

IF 0.6 3区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Discrete & Computational Geometry Pub Date : 2024-05-04 DOI:10.1007/s00454-024-00647-9

Andrii Arman, Andrii Bondarenko, Andriy Prymak

{"title":"具有指数照明数的恒宽凸面体","authors":"Andrii Arman, Andrii Bondarenko, Andriy Prymak","doi":"10.1007/s00454-024-00647-9","DOIUrl":null,"url":null,"abstract":"We show that there exist convex bodies of constant width in \\({\\mathbb {E}}^n\\) with illumination number at least \\((\\cos (\\pi /14)+o(1))^{-n}\\), answering a question by Kalai. Furthermore, we prove the existence of finite sets of diameter 1 in \\({\\mathbb {E}}^n\\) which cannot be covered by \\((2/\\sqrt{3}-o(1))^{n}\\) balls of diameter 1, improving a result of Bourgain and Lindenstrauss.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"118 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Convex Bodies of Constant Width with Exponential Illumination Number\",\"authors\":\"Andrii Arman, Andrii Bondarenko, Andriy Prymak\",\"doi\":\"10.1007/s00454-024-00647-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show that there exist convex bodies of constant width in \\\\({\\\\mathbb {E}}^n\\\\) with illumination number at least \\\\((\\\\cos (\\\\pi /14)+o(1))^{-n}\\\\), answering a question by Kalai. Furthermore, we prove the existence of finite sets of diameter 1 in \\\\({\\\\mathbb {E}}^n\\\\) which cannot be covered by \\\\((2/\\\\sqrt{3}-o(1))^{n}\\\\) balls of diameter 1, improving a result of Bourgain and Lindenstrauss.\",\"PeriodicalId\":50574,\"journal\":{\"name\":\"Discrete & Computational Geometry\",\"volume\":\"118 1\",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-05-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete & Computational Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00454-024-00647-9\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete & Computational Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00454-024-00647-9","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}

引用次数: 0

摘要

我们证明了在\({\mathbb {E}}^n\) 中存在照明数至少为 \((\cos (\pi /14)+o(1))^{-n}\)的恒宽凸体，这回答了卡莱提出的一个问题。此外，我们证明了在\({\mathbb {E}}^n\) 中存在直径为 1 的有限集合，这些集合不能被直径为 1 的球（（2/\sqrt{3}-o(1)）^{n}\）覆盖，从而改进了布尔甘和林登斯特劳斯的一个结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

Convex Bodies of Constant Width with Exponential Illumination Number

查看原文本刊更多论文

Convex Bodies of Constant Width with Exponential Illumination Number

We show that there exist convex bodies of constant width in \({\mathbb {E}}^n\) with illumination number at least \((\cos (\pi /14)+o(1))^{-n}\), answering a question by Kalai. Furthermore, we prove the existence of finite sets of diameter 1 in \({\mathbb {E}}^n\) which cannot be covered by \((2/\sqrt{3}-o(1))^{n}\) balls of diameter 1, improving a result of Bourgain and Lindenstrauss.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Discrete & Computational Geometry 数学-计算机：理论方法

CiteScore

1.80

自引率

12.50%

发文量

审稿时长

6-12 weeks

期刊介绍： Discrete & Computational Geometry (DCG) is an international journal of mathematics and computer science, covering a broad range of topics in which geometry plays a fundamental role. It publishes papers on such topics as configurations and arrangements, spatial subdivision, packing, covering, and tiling, geometric complexity, polytopes, point location, geometric probability, geometric range searching, combinatorial and computational topology, probabilistic techniques in computational geometry, geometric graphs, geometry of numbers, and motion planning.