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

Let Ω be an open convex set in Rm with finite width, and let vΩ be the torsion function for Ω, i.e. the solution of −Δv=1,v∈H10(Ω). An upper bound is obtained for the product of ∥vΩ∥L∞(Ω)λ(Ω), where λ(Ω) is the bottom of the spectrum of the Dirichlet Laplacian acting in L2(Ω). 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 ∥vΩ∥L1(Ω)λ(Ω)≥π224, and that this bound is sharp