Besicovitch-Morse type covering lemmas in metric spaces

TL;DR

This paper generalizes Besicovitch-Morse covering lemmas to complete Riemannian manifolds, explores conditions for BCP and WBCP equivalence, and investigates optimal constants in Euclidean spaces, providing new proofs and insights.

Contribution

It extends covering lemmas to broader spaces, analyzes properties like BCP and WBCP, and improves understanding of constants in Euclidean covering problems.

Findings

Generalized Besicovitch-Morse lemmas to Riemannian manifolds

Identified conditions for BCP and WBCP equivalence

Provided a new proof of the one-dimensional Besicovitch covering theorem

Abstract

The aims of this article is to generalize some useful Besicovitch-Morse type covering lemmas in complete Riemannian manifolds and try to find more spaces such that the so-called BCP and WBCP are equivalent while these two properties are weaker and still useful. We also get interest in the best constants of Besicovitch-type covering properties in Euclidean spaces and sorted out the best results of related problems before giving a new proof of Besicovitch covering theorem in the one-dimensional case.