The (largest) Lebesgue number and its relative version

TL;DR

This paper explores various definitions of the Lebesgue number for covers in metric spaces, introduces a relative version, and corrects a key lemma involving these concepts and quasi-homothetic maps.

Contribution

It introduces a relative Lebesgue number concept and provides a corrected proof of a lemma relating Lebesgue numbers and quasi-homothetic maps.

Findings

Comparison of different Lebesgue number definitions

Introduction of the relative Lebesgue number concept

Correction of Lemma 3.4 from Buyalo and Lebedeva's work

Abstract

In this paper we compare different definitions of the (largest) Lebesgue number of a cover $\mathcal{U}$ for a metric space $X$. We also introduce the relative version for the Lebesgue number of a covering family $\mathcal{U}$ for a subset $A\subseteq X$, and justify the relevance of introducing it by giving a corrected statement and proof of the Lemma 3.4 from S. Buyalo - N. Lebedeva paper "Dimensions of locally and asymptotically self-similar spaces", involving $\lambda$-quasi homothetic maps with coefficient $R$ between metric spaces, and the comparison of the mesh and the Lebesgue number of a covering family for a subset on both sides of the map.