On asymptotic Lebesgue's universal covering problem

IF 1 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2026-10-07 DOI:10.1112/blms.70514

A. Arman, A. Bondarenko, A. Prymak, D. Radchenko

{"title":"On asymptotic Lebesgue's universal covering problem","authors":"A. Arman, A. Bondarenko, A. Prymak, D. Radchenko","doi":"10.1112/blms.70514","DOIUrl":null,"url":null,"abstract":"A universal cover in <math>\n <semantics>\n <msup>\n <mi>E</mi>\n <mi>n</mi>\n </msup>\n <annotation>$\\mathbb {E}^{n}$</annotation>\n </semantics></math> is a measurable set that contains a congruent copy of any set of diameter 1. Lebesgue's universal covering problem, posed in 1914, asks for the convex set of smallest area that serves as a universal cover in the plane (<math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>=</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$n=2$</annotation>\n </semantics></math>). A simple universal cover in <math>\n <semantics>\n <msup>\n <mi>E</mi>\n <mi>n</mi>\n </msup>\n <annotation>$\\mathbb {E}^n$</annotation>\n </semantics></math> is provided by the classical theorem of Jung, which states that any set of diameter 1 in an <math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math>-dimensional Euclidean space is contained in a ball <math>\n <semantics>\n <msub>\n <mi>J</mi>\n <mi>n</mi>\n </msub>\n <annotation>$J_n$</annotation>\n </semantics></math> of radius <math>\n <semantics>\n <msqrt>\n <mstyle>\n <mfrac>\n <mi>n</mi>\n <mrow>\n <mn>2</mn>\n <mi>n</mi>\n <mo>+</mo>\n <mn>2</mn>\n </mrow>\n </mfrac>\n </mstyle>\n </msqrt>\n <annotation>$\\sqrt {\\tfrac{n}{2n+2}}$</annotation>\n </semantics></math>; in other words, <math>\n <semantics>\n <msub>\n <mi>J</mi>\n <mi>n</mi>\n </msub>\n <annotation>$J_n$</annotation>\n </semantics></math> is a universal cover in <math>\n <semantics>\n <msup>\n <mi>E</mi>\n <mi>n</mi>\n </msup>\n <annotation>$\\mathbb {E}^n$</annotation>\n </semantics></math>. We show that in high dimensions, Jung's ball <math>\n <semantics>\n <msub>\n <mi>J</mi>\n <mi>n</mi>\n </msub>\n <annotation>$J_n$</annotation>\n </semantics></math> is asymptotically optimal with respect to the volume, namely, for any universal cover <math>\n <semantics>\n <mrow>\n <mi>U</mi>\n <mo>⊂</mo>\n <msup>\n <mi>E</mi>\n <mi>n</mi>\n </msup>\n </mrow>\n <annotation>$U \\subset \\mathbb {E}^n$</annotation>\n </semantics></math>,\n\n ","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 10","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2026-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70514","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/blms.70514","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

A universal cover in $E^{n}$ is a measurable set that contains a congruent copy of any set of diameter 1. Lebesgue's universal covering problem, posed in 1914, asks for the convex set of smallest area that serves as a universal cover in the plane ( $n = 2$ ). A simple universal cover in $E^{n}$ is provided by the classical theorem of Jung, which states that any set of diameter 1 in an $n$ -dimensional Euclidean space is contained in a ball $J_{n}$ of radius $\sqrt{\frac{n}{2 n + 2}}$ ; in other words, $J_{n}$ is a universal cover in $E^{n}$ . We show that in high dimensions, Jung's ball $J_{n}$ is asymptotically optimal with respect to the volume, namely, for any universal cover $U \subset E^{n}$ ,

查看原文本刊更多论文

关于渐近Lebesgue的全称覆盖问题

在en $\mathbb {E}^{n}$中的全称覆盖是一个可测集合，它包含任何直径为1的集合的同余副本。Lebesgue的全称覆盖问题，提出于1914年，要求在平面（n=2$ n=2$）上作为全称覆盖的最小面积凸集。一个简单的n$ \mathbb {E}^n$的全称覆盖由Jung的经典定理提供，在n维欧几里得空间中任何直径为1的集合都包含在半径为n2的球J n J_n中N +2 $\sqrt {\tfrac{N}{2n+2}}$；换句话说，J n$ J_n$是E n$ \mathbb {E}^n$中的全称覆盖。我们证明了在高维空间中，Jung球jn $J_n$相对于体积是渐近最优的，即：对于任何全称覆盖U∧n$ U \子集\mathbb {E}^n$，

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊