Equivalence between validity of the $p$-Poincaré inequality and finiteness of the strict $p$-capacitary inradius

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