{"title":"Moduli of uniform convexity for convex sets","authors":"Carlo Alberto De Bernardi, Libor Veselý","doi":"10.1007/s00013-024-02031-8","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>C</i> be a proper, closed subset with nonempty interior in a normed space <i>X</i>. We define four variants of modulus of convexity for <i>C</i> and prove that they all coincide. This result, which is classical and well-known for <span>\\(C=B_X\\)</span> (the unit ball of <i>X</i>), requires a less easy proof than the particular case of <span>\\(B_X.\\)</span> We also show that if the modulus of convexity of <i>C</i> is not identically null, then <i>C</i> is bounded. This extends a result by M.V. Balashov and D. Repovš.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"123 4","pages":"413 - 422"},"PeriodicalIF":0.5000,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02031-8.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archiv der Mathematik","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00013-024-02031-8","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
Let C be a proper, closed subset with nonempty interior in a normed space X. We define four variants of modulus of convexity for C and prove that they all coincide. This result, which is classical and well-known for \(C=B_X\) (the unit ball of X), requires a less easy proof than the particular case of \(B_X.\) We also show that if the modulus of convexity of C is not identically null, then C is bounded. This extends a result by M.V. Balashov and D. Repovš.
我们为 C 定义了四种凸性模的变体，并证明它们都是重合的。对于 \(C=B_X\)（X 的单位球）来说，这个结果是经典且众所周知的，与 \(B_X.\)的特殊情况相比，这个结果的证明并不那么容易。 我们还证明了，如果 C 的凸模不等同于空，那么 C 是有界的。这扩展了 M.V. Balashov 和 D. Repovš 的一个结果。
期刊介绍：
Archiv der Mathematik (AdM) publishes short high quality research papers in every area of mathematics which are not overly technical in nature and addressed to a broad readership.
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