{"title":"A Note on Borsuk’s Problem in Minkowski Spaces","authors":"A. M. Raigorodskii, A. Sagdeev","doi":"10.1134/S1064562424701849","DOIUrl":null,"url":null,"abstract":"<p>In 1993, Kahn and Kalai famously constructed a sequence of finite sets in <i>d</i>-dimensional Euclidean spaces that cannot be partitioned into less than <span>\\({{(1.203 \\ldots + o(1))}^{{\\sqrt d }}}\\)</span> parts of smaller diameter. Their method works not only for the Euclidean, but for all <span>\\({{\\ell }_{p}}\\)</span>-spaces as well. In this short note, we observe that the larger the value of <i>p</i>, the stronger this construction becomes.</p>","PeriodicalId":531,"journal":{"name":"Doklady Mathematics","volume":"109 1","pages":"80 - 83"},"PeriodicalIF":0.5000,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Doklady Mathematics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1134/S1064562424701849","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In 1993, Kahn and Kalai famously constructed a sequence of finite sets in d-dimensional Euclidean spaces that cannot be partitioned into less than \({{(1.203 \ldots + o(1))}^{{\sqrt d }}}\) parts of smaller diameter. Their method works not only for the Euclidean, but for all \({{\ell }_{p}}\)-spaces as well. In this short note, we observe that the larger the value of p, the stronger this construction becomes.
期刊介绍：
Doklady Mathematics is a journal of the Presidium of the Russian Academy of Sciences. It contains English translations of papers published in Doklady Akademii Nauk (Proceedings of the Russian Academy of Sciences), which was founded in 1933 and is published 36 times a year. Doklady Mathematics includes the materials from the following areas: mathematics, mathematical physics, computer science, control theory, and computers. It publishes brief scientific reports on previously unpublished significant new research in mathematics and its applications. The main contributors to the journal are Members of the RAS, Corresponding Members of the RAS, and scientists from the former Soviet Union and other foreign countries. Among the contributors are the outstanding Russian mathematicians.
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