On principal frequencies, volume and inradius in convex sets

TL;DR

This paper establishes precise bounds for Poincaré-Sobolev constants in convex sets, linking geometric properties like inradius and volume, thereby unifying previous results on eigenvalues and torsional rigidity.

Contribution

It provides a unified, sharp double-sided estimate for Poincaré-Sobolev constants in convex sets based on inradius and measure, extending prior individual results.

Findings

Sharp bounds for Poincaré-Sobolev constants in convex sets

Unified approach encompassing eigenvalues and torsional rigidity

Extension of previous results by Hersch, Protter, Makai, Pólya, and Szegő

Abstract

We provide a sharp double-sided estimate for Poincar\'e-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, P\'olya and Szeg\H{o} (for the torsional rigidity), by means of a single proof.