{"title":"距离球的定量矫直","authors":"G. David, McKenna Kaczanowski, D. Pinkerton","doi":"10.14321/realanalexch.48.1.1626760923","DOIUrl":null,"url":null,"abstract":"We study\"distance spheres\": the set of points lying at constant distance from a fixed arbitrary subset $K$ of $[0,1]^d$. We show that, away from the regions where $K$ is\"too dense\"and a set of small volume, we can decompose $[0,1]^d$ into a finite number of sets on which the distance spheres can be\"straightened\"into subsets of parallel $(d-1)$-dimensional planes by a bi-Lipschitz map. Importantly, the number of sets and the bi-Lipschitz constants are independent of the set $K$.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.1000,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Quantitative Straightening of Distance Spheres\",\"authors\":\"G. David, McKenna Kaczanowski, D. Pinkerton\",\"doi\":\"10.14321/realanalexch.48.1.1626760923\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study\\\"distance spheres\\\": the set of points lying at constant distance from a fixed arbitrary subset $K$ of $[0,1]^d$. We show that, away from the regions where $K$ is\\\"too dense\\\"and a set of small volume, we can decompose $[0,1]^d$ into a finite number of sets on which the distance spheres can be\\\"straightened\\\"into subsets of parallel $(d-1)$-dimensional planes by a bi-Lipschitz map. Importantly, the number of sets and the bi-Lipschitz constants are independent of the set $K$.\",\"PeriodicalId\":44674,\"journal\":{\"name\":\"Real Analysis Exchange\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.1000,\"publicationDate\":\"2021-07-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Real Analysis Exchange\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.14321/realanalexch.48.1.1626760923\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Real Analysis Exchange","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.14321/realanalexch.48.1.1626760923","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
We study"distance spheres": the set of points lying at constant distance from a fixed arbitrary subset $K$ of $[0,1]^d$. We show that, away from the regions where $K$ is"too dense"and a set of small volume, we can decompose $[0,1]^d$ into a finite number of sets on which the distance spheres can be"straightened"into subsets of parallel $(d-1)$-dimensional planes by a bi-Lipschitz map. Importantly, the number of sets and the bi-Lipschitz constants are independent of the set $K$.
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