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arXiv · 2401.02762

Poincaré inequality and energy of separating sets

Emanuele Caputo·Nicola Cavallucci
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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.

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Emanuele Caputo, Nicola Cavallucci. 2024-01-05. Poincaré inequality and energy of separating sets. https://arxiv.org/abs/2401.02762

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