On a Pólya functional for rhombi, isosceles triangles, and thinning convex sets

Abstract

Let $\Omega$ be an open convex set in ${\mathbb{R}}^m$ with finite width, and with boundary $\partial \Omega$. Let $v_{\Omega }$ be the torsion function for $\Omega$, i.e., the solution of $-\Delta v=1,v{|}_{\partial \Omega }=0$. An upper bound is obtained for the product of $\Vert v_{\Omega }{\Vert }_{L^{\infty }(\Omega )}\lambda (\Omega )$, where $\lambda (\Omega )$ is the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(\Omega )$. The upper bound is sharp in the limit of a thinning sequence of convex sets. For planar rhombi and isosceles triangles with area $1$, it is shown that $\Vert v_{\Omega }{\Vert }_{L^1(\Omega )}\lambda (\Omega )\ge {\pi }^2/24$, and that this bound is sharp.