Capacitary inradius and Poincaré-Sobolev inequalities

We prove a two-sided estimate on the sharp Lp Poincaré 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 Poincaré-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.