Porosity and regularity in metric measure spaces

TL;DR

This paper characterizes uniformly porous sets in s-regular metric spaces using regular sets, providing an outline of the proof and an alternative proof for a key lemma, advancing understanding of porosity and regularity.

Contribution

It offers a new characterization of uniformly porous sets in s-regular metric spaces through the use of t-regular sets, with an alternative proof for a key lemma.

Findings

Uniformly porous sets are characterized by the existence of t-regular supersets.

An outline of the main proof strategy is provided.

An alternative proof for a crucial lemma is presented.

Abstract

This is a report of a joint work with E. J\"arvenp\"a\"a, M. J\"arvenp\"a\"a, T. Rajala, S. Rogovin, and V. Suomala. In [3], we characterized uniformly porous sets in $s$-regular metric spaces in terms of regular sets by verifying that a set $A$ is uniformly porous if and only if there is $t < s$ and a $t$-regular set $F \supset A$. Here we outline the main idea of the proof and also present an alternative proof for the crucial lemma needed in the proof of the result.