{"title":"关于闵科夫斯基空间中的博尔苏克问题的说明","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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-04-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1134/S1064562424701849\",\"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://link.springer.com/article/10.1134/S1064562424701849","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
摘要1993年，卡恩和卡莱在d维欧几里得空间中构建了一个著名的有限集序列，它不能被分割成直径小于({{(1.203 \ldots + o(1))}^{{\sqrt d }}}\) 的部分。他们的方法不仅适用于欧几里得空间，也适用于所有 \({{\ell }_{p}}\)-空间。在这篇短文中，我们观察到 p 的值越大，这种构造就越强。
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.
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