Accepted Paper

Inserted: 8 jan 2024
Last Updated: 12 feb 2025

Journal: Advances in Calculus of Variations
Year: 2024

37 pages, 1 figure.

Abstract:

We study geometric characterizations of the Poincaré inequality in doubling metric measure spaces in terms of properties of separating sets. Given a couple of points and a set separating them, such properties are formulated in terms of several possible notions of energy of the boundary, involving for instance the perimeter, codimension type Hausdorff measures, capacity, Minkowski content and approximate modulus of suitable families of curves. We prove the equivalence within each of these conditions and the 1-Poincaré inequality.