How to take shortcuts in Euclidean space: making a given set into a short quasi-convex set
type: Item
instance of: scholarly article
title: How to take shortcuts in Euclidean space: making a given set into a short quasi-convex set
zbMATH Open document ID: 1250.28001
author: Jonas Azzam
author: Raanan Schul
publication date: 2012-08-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/0912.1356
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 26B25
zbMATH DE Number: 6073825
zbMATH Keywords: quasi-convexity
zbMATH Keywords: 1-dimensional Haussdorff measure
zbMATH Keywords: traveling salesman problem
MaRDI profile type: Publication
OpenAlex ID: W2029529119
arXiv ID: 0912.1356
DOI: 10.1112/PLMS/PDS005
published in: Proceedings of the London Mathematical Society. Third Series
review text: A set \(\Gamma\subset \mathbb{R}^d\) is called \(C-\)quasi-convex if for any \(x,y\in\Gamma\) there is a path \(\gamma\subset\Gamma\) connecting \(x\) and \(y\) such that the length of \(\gamma\) does not exceed \(C|x-y|.\) The main result of the paper is the following one.NEWLINENEWLINETheorem. There exist an absolute constant \(C_1\) and depending on an integer \(d\geq 2\) a constant \(C_2 >1\) such that any set \(K\subset\mathbb{R}^d\) can be covered by a \(C_{2}-\)quasi-convex connected set \(\tilde{\Gamma}\) satisfying the inequality \({\mathbb{H}}^{1}(\tilde{\Gamma})\leq C_{1}{\mathbb{H}}^{1}(\Gamma)\) for any connected \(\Gamma\supset K.\) The symbol \({\mathbb{H}}^1\) stands here for the 1-dimensional Hausdorff measure.NEWLINENEWLINEFor \(d=2\) this result was obtained by \textit{P. W. Jones} [Invent. Math. 102, No. 1, 1--15 (1990; Zbl 0731.30018)].NEWLINENEWLINEThe authors obtain also a certain condition for the independence of the constant \(C_2\) of dimension \(d\).
reviewed by: Boris A. Kats
description: scientific article; zbMATH DE number 6073825