{"title":"关于二义轨迹的一个旧定理Erdős","authors":"P. Hajłasz","doi":"10.4064/cm8460-9-2021","DOIUrl":null,"url":null,"abstract":". Erdős proved in 1946 that if a set E ⊂ R n is closed and non-empty, then the set, called ambiguous locus or medial axis, of points in R n with the property that the nearest point in E is not unique, can be covered by countably many surfaces, each of finite ( n − 1) -dimensional measure. We improve the result by obtaining a new regularity result for these surfaces in terms of convexity and C 2 regularity.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":"{\"title\":\"On an old theorem of Erdős about ambiguous locus\",\"authors\":\"P. Hajłasz\",\"doi\":\"10.4064/cm8460-9-2021\",\"DOIUrl\":null,\"url\":null,\"abstract\":\". Erdős proved in 1946 that if a set E ⊂ R n is closed and non-empty, then the set, called ambiguous locus or medial axis, of points in R n with the property that the nearest point in E is not unique, can be covered by countably many surfaces, each of finite ( n − 1) -dimensional measure. We improve the result by obtaining a new regularity result for these surfaces in terms of convexity and C 2 regularity.\",\"PeriodicalId\":49216,\"journal\":{\"name\":\"Colloquium Mathematicum\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2022-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Colloquium Mathematicum\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4064/cm8460-9-2021\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Colloquium Mathematicum","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4064/cm8460-9-2021","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
. Erdős proved in 1946 that if a set E ⊂ R n is closed and non-empty, then the set, called ambiguous locus or medial axis, of points in R n with the property that the nearest point in E is not unique, can be covered by countably many surfaces, each of finite ( n − 1) -dimensional measure. We improve the result by obtaining a new regularity result for these surfaces in terms of convexity and C 2 regularity.
期刊介绍：
Colloquium Mathematicum is a journal devoted to the publication of original papers of moderate length addressed to a broad mathematical audience. It publishes results of original research, interesting new proofs of important theorems and research-expository papers in all fields of pure mathematics.
Two issues constitute a volume, and at least four volumes are published each year.
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