On Blaschke–Santaló diagrams for the torsional rigidity and the first Dirichlet eigenvalue

Abstract

The Pál inequality is a classical result which asserts that among all planar convex sets of given width the equilateral triangle is the one of minimal area. In this paper we prove three quantitative versions of this inequality, by quantifying how the closeness of the area of a convex set, of certain width, to the minimal value implies its closeness to the equilateral triangle. As a by-product, we also present a novel result concerning a quantitative inequality for the inradius of a set, under minimal width constraint.

On relations between principal eigenvalue and torsional rigidity

“…We obtain the full description of the Blaschke-Santaló diagram only for d = 1, while for d > 1 we only provide some bounds. Further properties of the Blaschke-Santaló diagram for λ(Ω) and T (Ω) are investigated in [22].…”

On a Pólya’s inequality for planar convex sets

“…• We recall that the infimum of the functional λ 1 (Ω)T (Ω) |Ω| on the class of open sets is zero (see [2,Remark 2.4]. As for the class of bounded convex subsets of R n (with n ≥ 2), it is conjectured that the infimum is given by the constant We note that this diagram has been theoretically studied in [10] and we refer to [1,7] for results in the case of open sets. In Figure 2, we plot an approximation of the diagram D obtained by randomly generating 10 5 convex polygons (we used the algorithm presented in [14]) for which we compute the involved functionals, we also plot the curves corresponding to the best known inequalities relating the 3 functionals, namely:…”

A Blaschke–Lebesgue theorem for the Cheeger constant

“…First used by Santaló in [18], this approach has now become a standard tool in shape optimization. We cite, e.g., [2,3,6,7,8,11,12,16], in which shape functionals of spectral and geometric type are studied. As it appears from the literature, the theoretical analysis, even if very fine, is in general not enough for an accurate description of the diagram.…”