The Besicovitch covering theorem for parabolic balls in Euclidean space
type: Item
instance of: scholarly article
title: The Besicovitch covering theorem for parabolic balls in Euclidean space
zbMATH Open document ID: 1417.52025
author: Tsubasa Itoh
published in: Hiroshima Mathematical Journal
publication date: 2019-03-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.hmj/1544238028
review text: The parabolic metric for $x=(x_1,\ldots,x_n),y=(y_1,\ldots,y_n) \in \mathbb{R}^n$, $n \ge 2$, is defined by $d(x,y)=\max\left\{\sqrt{(x_1-y_1)^2+\ldots+(x_{n-1}-y_{n-1})^2}, \sqrt{|x_n-y_n|}\right\}$. The corresponding parabolic ball with center $x \in \mathbb{R}^n$ and radius $r > 0$ is given by $B(x,r)=\{y \in \mathbb{R}^n: d(x,y) \le r\}$. The following variant of Besicovitch's covering theorem is shown: there is a positive integer $N=N(n)$ such that, for any family $\mathcal{F}$ of parabolic balls in $\mathbb{R}^n$ with uniformly bounded radii, there are $N$ subfamilies $\mathcal{G}_1,\ldots,\mathcal{G}_N \subseteq \mathcal{F}$ such that (i) each $\mathcal{G}_j$ consists of at most countably many disjoint balls and (ii) the set of all centers of balls from $\mathcal{F}$ is covered by $\mathcal{G}_1 \cup \ldots \cup \mathcal{G}_N$.
reviewed by: Christian Richter
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 05B40
Mathematics Subject Classification ID: 28A75
zbMATH DE Number: 7032358
zbMATH Keywords: Besicovitch covering theorem
zbMATH Keywords: parabolic ball
MaRDI profile type: Publication
OpenAlex ID: W2904795978
Wikidata QID: Q128764322
DOI: 10.32917/HMJ/1544238028
description: scientific article; zbMATH DE number 7032358