Banach空间I中可整流曲线的子集:旅行推销员定理中的锐指数

IF 0.6 Q3 MATHEMATICS

Illinois Journal of Mathematics Pub Date : 2020-02-27 DOI:10.1215/00192082-10592363

Matthew Badger, Sean McCurdy

{"title":"Banach空间I中可整流曲线的子集:旅行推销员定理中的锐指数","authors":"Matthew Badger, Sean McCurdy","doi":"10.1215/00192082-10592363","DOIUrl":null,"url":null,"abstract":"The Analyst's Traveling Salesman Problem is to find a characterization of subsets of rectifiable curves in a metric space. This problem was introduced and solved in the plane by Jones in 1990 and subsequently solved in higher-dimensional Euclidean spaces by Okikiolu in 1992 and in the infinite-dimensional Hilbert space $\\ell_2$ by Schul in 2007. In this paper, we establish sharp extensions of Schul's necessary and sufficient conditions for a bounded set $E\\subset\\ell_p$ to be contained in a rectifiable curve from $p=2$ to $1<p<\\infty$. While the necessary and sufficient conditions coincide when $p=2$, we demonstrate that there is a strict gap between the necessary condition and sufficient condition when $p\\neq 2$. We also identify and correct technical errors in the proof by Schul. This investigation is partly motivated by recent work of Edelen, Naber, and Valtorta on Reifenberg-type theorems in Banach spaces and complements work of Hahlomaa and recent work of David and Schul on the Analyst's TSP in general metric spaces.","PeriodicalId":56298,"journal":{"name":"Illinois Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"Subsets of rectifiable curves in Banach spaces I: Sharp exponents in traveling salesman theorems\",\"authors\":\"Matthew Badger, Sean McCurdy\",\"doi\":\"10.1215/00192082-10592363\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The Analyst's Traveling Salesman Problem is to find a characterization of subsets of rectifiable curves in a metric space. This problem was introduced and solved in the plane by Jones in 1990 and subsequently solved in higher-dimensional Euclidean spaces by Okikiolu in 1992 and in the infinite-dimensional Hilbert space $\\\\ell_2$ by Schul in 2007. In this paper, we establish sharp extensions of Schul's necessary and sufficient conditions for a bounded set $E\\\\subset\\\\ell_p$ to be contained in a rectifiable curve from $p=2$ to $1<p<\\\\infty$. While the necessary and sufficient conditions coincide when $p=2$, we demonstrate that there is a strict gap between the necessary condition and sufficient condition when $p\\\\neq 2$. We also identify and correct technical errors in the proof by Schul. This investigation is partly motivated by recent work of Edelen, Naber, and Valtorta on Reifenberg-type theorems in Banach spaces and complements work of Hahlomaa and recent work of David and Schul on the Analyst's TSP in general metric spaces.\",\"PeriodicalId\":56298,\"journal\":{\"name\":\"Illinois Journal of Mathematics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2020-02-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Illinois Journal of Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1215/00192082-10592363\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Illinois Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1215/00192082-10592363","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 5

摘要

分析员的旅行推销员问题是找到度量空间中可直曲线子集的特征。Jones于1990年在平面中引入并求解了这个问题，随后Okikiolu于1992年在高维欧几里得空间中求解了这个，Schul于2007年在无限维Hilbert空间$\ell_2$中求解了它。在本文中，我们建立了Schul关于有界集$E\subet\ell_p$包含在从$p=2$到$1本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Subsets of rectifiable curves in Banach spaces I: Sharp exponents in traveling salesman theorems

The Analyst's Traveling Salesman Problem is to find a characterization of subsets of rectifiable curves in a metric space. This problem was introduced and solved in the plane by Jones in 1990 and subsequently solved in higher-dimensional Euclidean spaces by Okikiolu in 1992 and in the infinite-dimensional Hilbert space $\ell_2$ by Schul in 2007. In this paper, we establish sharp extensions of Schul's necessary and sufficient conditions for a bounded set $E\subset\ell_p$ to be contained in a rectifiable curve from $p=2$ to $1

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Illinois Journal of Mathematics 数学-数学

CiteScore

0.90

自引率

0.00%

发文量

期刊介绍： IJM strives to publish high quality research papers in all areas of mainstream mathematics that are of interest to a substantial number of its readers. IJM is published by Duke University Press on behalf of the Department of Mathematics at the University of Illinois at Urbana-Champaign.