On the optimal sets in Pólya and Makai type inequalities

TL;DR

For bounded convex sets $\Omega\subset\mathbb{R}^n$, the paper develops a quantitative framework to study shape functionals built from the torsional rigidity $T(\Omega)$ and the first Dirichlet eigenvalue $\lambda(\Omega)$, where no optimizer exists. It introduces reminder terms $\alpha(\Omega)$ and $\beta(\Omega)$ to connect the geometry of thinning domains to stability of the classical inequalities, using inner parallel sets, web torsion $W(\Omega)$, and quermassintegrals. The authors prove three main quantitative results: (i) a sharp $n$-dimensional bound for the Pólya torsion functional in terms of $\beta(\Omega)$ with optimal $\alpha(\Omega)$-dependence; (ii) a sharp bound for the Pólya eigenvalue functional with similar dependence; (iii) a $2$-D Makai bound and Hersch-type estimates, together with a geometric characterization of minimizing sequences as thinning domains. They also derive corollaries relating $\alpha(\Omega)$ to the main functionals and provide sharpness results and counterexamples, establishing a comprehensive geometric stability theory for these isoperimetric-type inequalities.

Abstract

In this paper, we examine some shape functionals, introduced by Pólya 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.

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Theorems & Definitions (29)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Proposition 1.6
• Corollary 1.7
• Definition 2.1
• Definition 2.2
• Definition 2.3