On the sharp Makai inequality

TL;DR

This paper extends the Makai inequality to convex sets in higher dimensions, providing bounds for Poincaré-Sobolev constants based on distance function norms, and compares these with other classes of sets.

Contribution

It generalizes Makai's planar result to higher dimensions and different set classes, offering new bounds and an alternative proof of the Hersch-Protter inequality.

Findings

Bounds for Poincaré-Sobolev constants in convex sets

Comparison of Makai constants across set classes

Alternative proof of Hersch-Protter inequality

Abstract

On a convex bounded open set, we prove that Poincar\'e-Sobolev constants for functions vanishing at the boundary can be bounded from below in terms of the norm of the distance function in a suitable Lebesgue space. This generalizes a result shown, in the planar case, by E. Makai, for the torsional rigidity. In addition, we compare the sharp Makai constants obtained in the class of convex sets with the optimal constants defined in other classes of open sets. Finally, an alternative proof of the Hersch-Protter inequality for convex sets is given.