On principal frequencies, volume and inradius in convex sets

We provide a sharp double-sided estimate for Poincare–Sobolev constants on a convex set, in terms of its inradius and $$N-$$dimensional measure. Our results extend and unify previous works by Hersch and Protter (for the first eigenvalue) and of Makai, Polya and Szegő (for the torsional rigidity), by means of a single proof.