The restricted content and the \(d\)-dimensional Analyst's travelling salesman theorem for general sets
type: Item
instance of: scholarly article
title: The restricted content and the \(d\)-dimensional Analyst's travelling salesman theorem for general sets
zbMATH Open document ID: 1503.28006
published in: Advances in Mathematics
publication date: 2022-02-11T00:00:00Z
review text: A multiscale geometric criterion for a set \(E \subset \mathbb{R}^2\) to be contained in a rectifiable curve has been proved by \textit{P. W. Jones} [Invent. Math. 102, No. 1, 1--15 (1990; Zbl 0731.30018)] who introduced \(\beta\)-numbers. A characterization of subsets of higher dimensional surfaces was studied by \textit{M. Villa} [``Higher dimensional Jordan curves'', Preprint, \url{arXiv:1908.10289}] with the use of a variant of Jones' \(\beta\)-numbers introduced by \textit{J. Azzam} and \textit{R. Schul} [Math. Ann. 370, No. 3--4, 1389--1476 (2018; Zbl 1388.28005)]. Villa considered lower regular subsets (with respect to a \(d\)-dimensional Hausdorff content) of a certain class of surfaces. In the first section of the paper the author has defined a new \(d\)-dimensional variant of Jones' \(\beta\)-number for any set in \(\mathbb{R}^n\) and \(1 \leqslant d < n\) (def. 1.11, def. 3.13 and remark 3.16): Definition 1.11. Let \(1 \leq p < \infty\), \(E \subseteq \mathbb{R}^n\), \(B\) be a ball centred on \(E\) and \(L\) a \(d\)-plane. Define \begin{align*} \beta^{d,p}_E(B,L)^p&=\frac{1}{r^d_B}\int\left(\frac{\operatorname{dis} (x,L)}{r_B}\right)^p d\mathscr{R}^d_{E\cap B}\\ &=\frac{p}{r^d_B} \int^1_0 \mathscr{R}^d_{E\cap B}(\{x\in E :\operatorname{dis} (x,L)>tr_B\})t^{p-1}dt, \end{align*} and \[ \beta^{d,p}_E(B)=\inf\{ \beta^{d,p}_E (B,L): \text{ is a \(d\)-plane} \}. \] The main results of the paper are Theorems 1.13 and 1.14 Theorem 1.13. Let \(1 \leq d < n\), \(C_0 > 1\) and \(1 \leq p < p(d)\). There exists a constant \(c_1 > 0\) such that the following holds. Suppose \(E \subseteq F \subseteq \mathbb{R}^n\), where \(F\) is \((c,d)\)-lower content regular for some \(c\geq c_1\). Let \(\mathscr{D}^E\) be a system of Christ-David cubes for \(E\), and let \(\mathscr{D}^F\) be an extension of \(\mathscr{D}^E\) for \(F\), as above. Let \(Q^E_0 \in \mathscr{D}^E\) and let \(Q^F_0\) be the cube in \(\mathscr{D}^F\) with the same centre and side length as \(Q^E_0\). The…
reviewed by: Władysław Wilczyński
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 28A12
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 7472318
zbMATH Keywords: rectifiability
zbMATH Keywords: travelling salesman theorem
zbMATH Keywords: beta numbers
zbMATH Keywords: Hausdorff content
author: Matthew R. Hyde
MaRDI profile type: Publication
full work available at URL: j.aim.2022.108189
OpenAlex ID: W4205513163
Wikidata QID: Q113880994
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DOI: 10.1016/J.AIM.2022.108189
description: scientific article; zbMATH DE number 7472318