Sharp Makai-type inequalities for the best Poincar\'e-Sobolev constants

TL;DR

This paper establishes sharp lower bounds for Poincaré-Sobolev constants in convex domains using geometric measures like perimeter and volume, generalizing classical results and introducing new bounds for distance functions.

Contribution

It provides explicit optimal constants for inequalities relating Poincaré-Sobolev constants to geometric properties, extending Makai's classical torsional rigidity result to higher dimensions.

Findings

Derived explicit lower bounds for Poincaré-Sobolev constants in convex sets.

Established new geometric bounds for Lebesgue norms of distance functions.

Unified sharp inequalities involving perimeter, inradius, and volume.

Abstract

Given a bounded convex open set $\Omega\subseteq \mathbb R^N$, we prove that the Poincar\'e-Sobolev constants $\lambda_{p,q}(\Omega)$ can be bounded from below by the $p$-power of the ratio between the perimeter of $\Omega$ and a suitable power of its volume, with an optimal constant which is explicitly given. This generalises an old result for torsional rigidity due to Makai when $N=2$. The proof relies on new geometric optimal bounds for the Lebesgue norms of the distance function from the boundary which are of independent interest. These results allow us to give a complete picture of the sharp inequalities for $\lambda_{p,q}(\Omega)$ in terms of suitable powers of perimeter, inradius and volume of $\Omega$.