{"title":"分数超容积不等式的相等条件","authors":"Mark Meyer","doi":"10.1007/s00454-024-00672-8","DOIUrl":null,"url":null,"abstract":"<p>While studying set function properties of Lebesgue measure, F. Barthe and M. Madiman proved that Lebesgue measure is fractionally superadditive on compact sets in <span>\\(\\mathbb {R}^n\\)</span>. In doing this they proved a fractional generalization of the Brunn–Minkowski–Lyusternik (BML) inequality in dimension <span>\\(n=1\\)</span>. In this paper we will prove the equality conditions for the fractional superadditive volume inequalites for any dimension. The non-trivial equality conditions are as follows. In the one-dimensional case we will show that for a fractional partition <span>\\((\\mathcal {G},\\beta )\\)</span> and nonempty sets <span>\\(A_1,\\dots ,A_m\\subseteq \\mathbb {R}\\)</span>, equality holds iff for each <span>\\(S\\in \\mathcal {G}\\)</span>, the set <span>\\(\\sum _{i\\in S}A_i\\)</span> is an interval. In the case of dimension <span>\\(n\\ge 2\\)</span> we will show that equality can hold if and only if the set <span>\\(\\sum _{i=1}^{m}A_i\\)</span> has measure 0.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"43 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Equality Conditions for the Fractional Superadditive Volume Inequalities\",\"authors\":\"Mark Meyer\",\"doi\":\"10.1007/s00454-024-00672-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>While studying set function properties of Lebesgue measure, F. Barthe and M. Madiman proved that Lebesgue measure is fractionally superadditive on compact sets in <span>\\\\(\\\\mathbb {R}^n\\\\)</span>. In doing this they proved a fractional generalization of the Brunn–Minkowski–Lyusternik (BML) inequality in dimension <span>\\\\(n=1\\\\)</span>. In this paper we will prove the equality conditions for the fractional superadditive volume inequalites for any dimension. The non-trivial equality conditions are as follows. In the one-dimensional case we will show that for a fractional partition <span>\\\\((\\\\mathcal {G},\\\\beta )\\\\)</span> and nonempty sets <span>\\\\(A_1,\\\\dots ,A_m\\\\subseteq \\\\mathbb {R}\\\\)</span>, equality holds iff for each <span>\\\\(S\\\\in \\\\mathcal {G}\\\\)</span>, the set <span>\\\\(\\\\sum _{i\\\\in S}A_i\\\\)</span> is an interval. In the case of dimension <span>\\\\(n\\\\ge 2\\\\)</span> we will show that equality can hold if and only if the set <span>\\\\(\\\\sum _{i=1}^{m}A_i\\\\)</span> has measure 0.</p>\",\"PeriodicalId\":50574,\"journal\":{\"name\":\"Discrete & Computational Geometry\",\"volume\":\"43 1\",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-07-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete & Computational Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00454-024-00672-8\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete & Computational Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00454-024-00672-8","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Equality Conditions for the Fractional Superadditive Volume Inequalities
While studying set function properties of Lebesgue measure, F. Barthe and M. Madiman proved that Lebesgue measure is fractionally superadditive on compact sets in \(\mathbb {R}^n\). In doing this they proved a fractional generalization of the Brunn–Minkowski–Lyusternik (BML) inequality in dimension \(n=1\). In this paper we will prove the equality conditions for the fractional superadditive volume inequalites for any dimension. The non-trivial equality conditions are as follows. In the one-dimensional case we will show that for a fractional partition \((\mathcal {G},\beta )\) and nonempty sets \(A_1,\dots ,A_m\subseteq \mathbb {R}\), equality holds iff for each \(S\in \mathcal {G}\), the set \(\sum _{i\in S}A_i\) is an interval. In the case of dimension \(n\ge 2\) we will show that equality can hold if and only if the set \(\sum _{i=1}^{m}A_i\) has measure 0.
期刊介绍:
Discrete & Computational Geometry (DCG) is an international journal of mathematics and computer science, covering a broad range of topics in which geometry plays a fundamental role. It publishes papers on such topics as configurations and arrangements, spatial subdivision, packing, covering, and tiling, geometric complexity, polytopes, point location, geometric probability, geometric range searching, combinatorial and computational topology, probabilistic techniques in computational geometry, geometric graphs, geometry of numbers, and motion planning.