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

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.