Inserted: 27 aug 2018
Last Updated: 1 nov 2018
This paper evolved from a set of hand-written notes for a talk delivered during the conferences ''Variational and PDE problems in Geometric Analysis'' and ''Recent advances in Geometric Analysis'' held in June 2018 in Bologna and Pisa, respectively. The organizers Chiara Guidi & Vittorio Martino and Andrea Malchiodi & Luciano Mari are kindly acknowledged.
Dedicated to Michelino Brasco, master craftsman and father, on the occasion of his 70th birthday.
We generalize to the case of the $p-$Laplacian an old result by Hersch and Protter. Namely, we show that it is possible to estimate from below the first eigenvalue of the Dirichlet $p-$Laplacian of a convex set in terms of its inradius. We also prove a lower bound in terms of isoperimetric ratios and we briefly discuss the more general case of Poincar\'e-Sobolev embedding constants. Eventually, we highlight an open problem.
Keywords: Nonlinear eigenvalue problems, convex sets, Cheeger constant, Inradius, $p-$Laplacian