{"title":"论小维度中的巴拿赫-马祖尔距离","authors":"Tomasz Kobos, Marin Varivoda","doi":"10.1007/s00454-024-00641-1","DOIUrl":null,"url":null,"abstract":"<p>We establish some results on the Banach–Mazur distance in small dimensions. Specifically, we determine the Banach–Mazur distance between the cube and its dual (the cross-polytope) in <span>\\(\\mathbb {R}^3\\)</span> and <span>\\(\\mathbb {R}^4\\)</span>. In dimension three this distance is equal to <span>\\(\\frac{9}{5}\\)</span>, and in dimension four, it is equal to 2. These findings confirm well-known conjectures, which were based on numerical data. Additionally, in dimension two, we use the asymmetry constant to provide a geometric construction of a family of convex bodies that are equidistant to all symmetric convex bodies.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Banach–Mazur Distance in Small Dimensions\",\"authors\":\"Tomasz Kobos, Marin Varivoda\",\"doi\":\"10.1007/s00454-024-00641-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We establish some results on the Banach–Mazur distance in small dimensions. Specifically, we determine the Banach–Mazur distance between the cube and its dual (the cross-polytope) in <span>\\\\(\\\\mathbb {R}^3\\\\)</span> and <span>\\\\(\\\\mathbb {R}^4\\\\)</span>. In dimension three this distance is equal to <span>\\\\(\\\\frac{9}{5}\\\\)</span>, and in dimension four, it is equal to 2. These findings confirm well-known conjectures, which were based on numerical data. Additionally, in dimension two, we use the asymmetry constant to provide a geometric construction of a family of convex bodies that are equidistant to all symmetric convex bodies.</p>\",\"PeriodicalId\":50574,\"journal\":{\"name\":\"Discrete & Computational Geometry\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-04-13\",\"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/s00454-024-00641-1\",\"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/s00454-024-00641-1","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}
We establish some results on the Banach–Mazur distance in small dimensions. Specifically, we determine the Banach–Mazur distance between the cube and its dual (the cross-polytope) in \(\mathbb {R}^3\) and \(\mathbb {R}^4\). In dimension three this distance is equal to \(\frac{9}{5}\), and in dimension four, it is equal to 2. These findings confirm well-known conjectures, which were based on numerical data. Additionally, in dimension two, we use the asymmetry constant to provide a geometric construction of a family of convex bodies that are equidistant to all symmetric convex bodies.
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.
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