Gil Bor, L. Hernández-Lamoneda, Valent'in Jim'enez-Desantiago, Luis Montejano-Peimbert
{"title":"On the isometric conjecture of Banach","authors":"Gil Bor, L. Hernández-Lamoneda, Valent'in Jim'enez-Desantiago, Luis Montejano-Peimbert","doi":"10.2140/gt.2021.25.2621","DOIUrl":null,"url":null,"abstract":"Let $V$ be a Banach space where for fixed $n$, $1<n<\\dim(V)$, all of its $n$-dimensional subspaces are isometric. In 1932, Banach asked if under this hypothesis $V$ is necessarily a Hilbert space. Gromov, in 1967, answered it positively for even $n$ and all $V$. In this paper we give a positive answer for real $V$ and odd $n$ of the form $n=4k+1$, with the possible exception of $n=133.$ Our proof relies on a new characterization of ellipsoids in ${\\mathbb{R}}^n$, $n\\geq 5$, as the only symmetric convex bodies all of whose linear hyperplane sections are linearly equivalent affine bodies of revolution.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":2.0000,"publicationDate":"2019-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Geometry & Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/gt.2021.25.2621","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
期刊介绍:
Geometry and Topology is a fully refereed journal covering all of geometry and topology, broadly understood. G&T is published in electronic and print formats by Mathematical Sciences Publishers.
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