On the optimal sets in Pólya and Makai type inequalities

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.

💡 Research Summary

The paper studies shape functionals involving the torsional rigidity T(Ω) and the first Dirichlet eigenvalue λ(Ω) of bounded, open, convex subsets Ω of ℝⁿ (n ≥ 2). Classical results—Pólya’s inequality for torsion and eigenvalue, and Makai’s inequality for torsion and eigenvalue with the inradius—show that the optimal value is attained by a ball, but no minimizer exists within the class of convex sets; the infimum is approached by “thin” domains such as flattening rectangles, triangles, or cylinders. The authors aim to turn these qualitative statements into quantitative estimates that describe how close a domain must be to a slab when the functional is near its extremal value.

Two geometric remainder terms are introduced:

• α(Ω) = w_Ω·diam(Ω), where w_Ω is the minimal width (thickness) and diam(Ω) the diameter. α measures how “thinning” a domain is; α → 0 characterizes sequences of thinning domains.
• β(Ω) = P(Ω) R_Ω/|Ω| − 1, where P(Ω) is the perimeter, R_Ω the inradius, and |Ω| the volume. β is dimensionless and captures a different aspect of the geometry.

Proposition 1.1 establishes a lower bound β(Ω) ≥ C₀(n) α(Ω), with an explicit constant, and shows that the reverse inequality cannot hold in general (e.g., collapsing pyramids give β bounded away from zero while α → 0). Hence β provides weaker control than α.

The main quantitative results are three families of inequalities, each relating the deviation of a functional from its optimal value to powers of β(Ω). For a convex set Ω ∈ K_n:

1. Pólya torsion functional J₁(Ω) = T(Ω) P(Ω)²/|Ω|³ satisfies \