Equivalence between validity of the $p$-Poincar\'e inequality and finiteness of the strict $p$-capacitary inradius

Abstract

It is shown that the $p$-Poincar\'e inequality holds on an open set $\Omega$ in $\mathbb{R}^n$ if and only if the strict $p$-capacitary inradius of $\Omega$ is finite. To that end, new upper and lower bounds for the infimum for the associated nonlinear Rayleigh quotients are derived.