On the optimal sets in P\'olya and Makai type inequalities

TL;DR

This paper investigates shape functionals related to torsional rigidity and eigenvalues in convex sets, establishing new bounds and insights into the structure and optimality of these geometric quantities.

Contribution

It introduces new quantitative bounds for Pólya and Makai type inequalities, revealing properties of minimizing sequences and their geometric structure.

Findings

Established new bounds for shape functionals

Provided insights into the structure of optimal sets

Analyzed behavior of minimizing sequences

Abstract

In this paper, we examine some shape functionals, introduced by P\'olya and Makai, involving the torsional rigidity and the first Dirichlet-Laplacian eigenvalue for bounded, open and convex sets of $\mathbb{R}^n$. We establish new quantitative bounds, which give us key properties and information on the behavior of the optimizing sequences. In particular, we consider two kinds of reminder terms that provide information about the structure of these minimizing sequences, such as information about the thickness.