  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.
