{"title":"正基、锥体、赫利型定理","authors":"Imre Bárány","doi":"10.1515/ms-2024-0054","DOIUrl":null,"url":null,"abstract":"Assume that <jats:italic>k</jats:italic> ≤ <jats:italic>d</jats:italic> is a positive integer and 𝓒 is a finite collection of convex bodies in ℝ<jats:sup> <jats:italic>d</jats:italic> </jats:sup>. We prove a Helly-type theorem: If for every subfamily 𝓒<jats:sup>*</jats:sup> ⊂ 𝓒 of size at most max{<jats:italic>d</jats:italic> + 1, 2(<jats:italic>d</jats:italic> – <jats:italic>k</jats:italic> + 1)} the set ⋂ 𝓒<jats:sup>*</jats:sup> contains a <jats:italic>k</jats:italic>-dimensional cone, then so does ⋂ 𝓒. One ingredient in the proof is another Helly-type theorem about the dimension of lineality spaces of convex cones.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":"26 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Positive bases, cones, Helly-type theorems\",\"authors\":\"Imre Bárány\",\"doi\":\"10.1515/ms-2024-0054\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Assume that <jats:italic>k</jats:italic> ≤ <jats:italic>d</jats:italic> is a positive integer and 𝓒 is a finite collection of convex bodies in ℝ<jats:sup> <jats:italic>d</jats:italic> </jats:sup>. We prove a Helly-type theorem: If for every subfamily 𝓒<jats:sup>*</jats:sup> ⊂ 𝓒 of size at most max{<jats:italic>d</jats:italic> + 1, 2(<jats:italic>d</jats:italic> – <jats:italic>k</jats:italic> + 1)} the set ⋂ 𝓒<jats:sup>*</jats:sup> contains a <jats:italic>k</jats:italic>-dimensional cone, then so does ⋂ 𝓒. One ingredient in the proof is another Helly-type theorem about the dimension of lineality spaces of convex cones.\",\"PeriodicalId\":18282,\"journal\":{\"name\":\"Mathematica Slovaca\",\"volume\":\"26 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-07-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematica Slovaca\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/ms-2024-0054\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematica Slovaca","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/ms-2024-0054","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
假设 k ≤ d 是正整数，𝓒 是 ℝ d 中凸体的有限集合。我们将证明一个海尔利类型定理：如果对于每个大小至多为 max{d + 1, 2(d - k + 1)} 的子域 𝓒* ⊂ 𝓒 的集合 ⋂ 𝓒* 包含一个 k 维锥，那么 ⋂ 𝓒 也包含一个 k 维锥。证明的一个要素是另一个关于凸锥体线性空间维数的海利定理。
Assume that k ≤ d is a positive integer and 𝓒 is a finite collection of convex bodies in ℝd. We prove a Helly-type theorem: If for every subfamily 𝓒* ⊂ 𝓒 of size at most max{d + 1, 2(d – k + 1)} the set ⋂ 𝓒* contains a k-dimensional cone, then so does ⋂ 𝓒. One ingredient in the proof is another Helly-type theorem about the dimension of lineality spaces of convex cones.
期刊介绍：
Mathematica Slovaca, the oldest and best mathematical journal in Slovakia, was founded in 1951 at the Mathematical Institute of the Slovak Academy of Science, Bratislava. It covers practically all mathematical areas. As a respectful international mathematical journal, it publishes only highly nontrivial original articles with complete proofs by assuring a high quality reviewing process. Its reputation was approved by many outstanding mathematicians who already contributed to Math. Slovaca. It makes bridges among mathematics, physics, soft computing, cryptography, biology, economy, measuring, etc. The Journal publishes original articles with complete proofs. Besides short notes the journal publishes also surveys as well as some issues are focusing on a theme of current interest.
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