A Proof of a Conjecture on a Pólya Functional

Abstract

Let $λ_1(Ω)$ and $T(Ω)$ denote the first Dirichlet eigenvalue and torsional rigidity of a bounded convex domain $Ω\subset\mathbb{R}^2$, and let $M=\max_Ωu$, where $u$ is the torsion function. We prove the sharp inequalities $π^2/24<λ_1(Ω)T(Ω)/|Ω|<π^2/12$. This proves the two-dimensional case of Conjecture 4.2 proposed by van den Berg, Buttazzo, and Pratelli. Its planar formulation was later restated as Conjecture 1.1 by Bañuelos and Mariano, who proved it for triangles and rectangles. The lower bound follows by combining Payne's strict estimate for $λ_1M$ with the sharp torsion-efficiency inequality $T(Ω)\geq |Ω|M/3$. For smooth strictly convex domains, we use an Airy stress potential to construct a convex body whose surface-area measure is the torsional first-variation measure. A sharp one-dimensional inequality for directional maximum profiles then yields a stronger geometric containment. For the upper bound, we factor the Pólya functional through the second torsion moment. A level-set torsion--perimeter inequality gives the factor $5/6$, while a sharp weighted one-dimensional estimate gives the factor $π^2/10$. Collapsing triangles and elongating rectangles show that both endpoint constants, as well as the efficiency constant $1/3$, are sharp.