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

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