Capacitary inradius and Poincar\'e-Sobolev inequalities

We prove a two-sided estimate on the sharp $L^p$ 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.