{"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":"33 1","pages":""},"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}
引用次数: 6
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.
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.