A Geometric Approach to Poincaré Inequality and Minkowski Content of Separating Sets

Abstract The goal of this paper is to continue the study of the relation between the Poincaré inequality and the lower bounds of Minkowski content of separating sets, initiated in our previous work [5]. A new shorter proof is provided. It is based on the study of the lower bound of a new geometric quantity, called separating ratio. The main result in this work is the quantitative comparison, in the locally quasigeodesic case, of the infima of the separating ratio and the Minkowski content of separating sets. The proof is based on a novel approach: it uses a new function, called the position function, which allows to fibrate a set in boundaries of separating sets. No Poincaré assumption is needed to prove the main result. We also extend the proof to measure graphs, where due to the combinatorial nature of the problem, the approach is more intuitive. In the appendix, we revise some classical characterizations of the $p$-Poincaré inequality, showing that the equivalences remain true if the several conditions hold only for a fixed couple of points.