{"title":"Some sharp inequalities for norms in $$\\mathbb {R}^n$$ and $$\\mathbb {C}^n$$","authors":"Stefan Gerdjikov, Nikolai Nikolov","doi":"10.1007/s00605-024-02004-7","DOIUrl":null,"url":null,"abstract":"<p>The main result of this paper is that for any norm on a complex or real <i>n</i>-dimensional linear space, every extremal basis satisfies inverted triangle inequality with scaling factor <span>\\(2^n-1\\)</span>. Furthermore, the constant <span>\\(2^n-1\\)</span> is tight. We also prove that the norms of any two extremal bases are comparable with a factor of <span>\\(2^n-1\\)</span>, which, intuitively, means that any two extremal bases are quantitatively equivalent with the stated tolerance.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-024-02004-7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The main result of this paper is that for any norm on a complex or real n-dimensional linear space, every extremal basis satisfies inverted triangle inequality with scaling factor \(2^n-1\). Furthermore, the constant \(2^n-1\) is tight. We also prove that the norms of any two extremal bases are comparable with a factor of \(2^n-1\), which, intuitively, means that any two extremal bases are quantitatively equivalent with the stated tolerance.