Capacitary inradius and Poincar\'e-Sobolev inequalities

We prove a two-sided estimate on the sharp Lᵖ Poincar\'e constant of a general open set, in terms of a capacitary variant of its inradius. This extends a result by Maz'ya and Shubin, originally devised for the case p=2, in the subconformal regime. We cover the whole range of p, by allowing in particular the extremal cases p=1 (Cheeger's constant) and p=N (conformal case), as well. We also discuss the more general case of the sharp Poincar\'e-Sobolev embedding constants and get an analogous result. Finally, we present a brief discussion on the superconformal case, as well as some examples and counter-examples.