Geometric characterizations of ${\sf PI}$ spaces: an overview of some modern techniques

TL;DR

This paper surveys modern geometric techniques used to characterize metric measure spaces satisfying a Poincaré inequality, focusing on both one-dimensional and codimension-one objects, and discusses new strategies and open questions in the field.

Contribution

It provides an overview of recent characterizations of PI spaces using geometric objects and introduces new methods for constructing examples and understanding their properties.

Findings

Characterizations via pencil of curves and modulus estimates.

Use of relative isoperimetric inequalities and separating sets.

Connections between separating sets and obstacle-avoidance principles.

Abstract

We survey recent results on the study of metric measure spaces satisfying a Poincar\'e inequality. We overview recent characterizations in terms of objects of dimension 1, such as pencil of curves, modulus estimates and obstacle-avoidance principles. Then, we turn our attention to characterizations in terms of objects of codimension 1, such as relative isoperimetric inequalities and separating sets, the last one obtained in collaboration with N. Cavallucci in [arXiv:2401.02762]. We propose a strategy to provide examples using our characterization in the toy-model of the Euclidean case. We also discuss a more geometric relation between separating sets and obstacle-avoidance principles, obtained in [IMRN, Vol. 2025, Issue 1, Jan. 2025, rnae276]. Finally, we recall some open questions in the field.