{"title":"均匀凸巴拿赫空间中的无维赫里定理","authors":"G. Ivanov","doi":"arxiv-2409.05744","DOIUrl":null,"url":null,"abstract":"We study the ``no-dimensional'' analogue of Helly's theorem in Banach spaces.\nSpecifically, we obtain the following no-dimensional Helly-type results for\nuniformly convex Banach spaces: Helly's theorem, fractional Helly's theorem,\ncolorful Helly's theorem, and colorful fractional Helly's theorem. The combinatorial part of the proofs for these Helly-type results is\nidentical to the Euclidean case as presented in \\cite{adiprasito2020theorems}.\nThe primary difference lies in the use of a certain geometric inequality in\nplace of the Pythagorean theorem. This inequality can be explicitly expressed\nin terms of the modulus of convexity of a Banach space.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"58 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"No-dimensional Helly's theorem in uniformly convex Banach spaces\",\"authors\":\"G. Ivanov\",\"doi\":\"arxiv-2409.05744\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the ``no-dimensional'' analogue of Helly's theorem in Banach spaces.\\nSpecifically, we obtain the following no-dimensional Helly-type results for\\nuniformly convex Banach spaces: Helly's theorem, fractional Helly's theorem,\\ncolorful Helly's theorem, and colorful fractional Helly's theorem. The combinatorial part of the proofs for these Helly-type results is\\nidentical to the Euclidean case as presented in \\\\cite{adiprasito2020theorems}.\\nThe primary difference lies in the use of a certain geometric inequality in\\nplace of the Pythagorean theorem. This inequality can be explicitly expressed\\nin terms of the modulus of convexity of a Banach space.\",\"PeriodicalId\":501036,\"journal\":{\"name\":\"arXiv - MATH - Functional Analysis\",\"volume\":\"58 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Functional Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.05744\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Functional Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.05744","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
No-dimensional Helly's theorem in uniformly convex Banach spaces
We study the ``no-dimensional'' analogue of Helly's theorem in Banach spaces.
Specifically, we obtain the following no-dimensional Helly-type results for
uniformly convex Banach spaces: Helly's theorem, fractional Helly's theorem,
colorful Helly's theorem, and colorful fractional Helly's theorem. The combinatorial part of the proofs for these Helly-type results is
identical to the Euclidean case as presented in \cite{adiprasito2020theorems}.
The primary difference lies in the use of a certain geometric inequality in
place of the Pythagorean theorem. This inequality can be explicitly expressed
in terms of the modulus of convexity of a Banach space.