On principal frequencies, volume and inradius in convex sets

We provide a sharp double-sided estimate for Poincaré-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, Pólya and Szegő (for the torsional rigidity), by means of a single proof.

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