Blaschke-santaló diagram for volume, perimeter, and first dirichlet eigenvalue

We are interested in the study of Blaschke-Santaló diagrams describing the possible inequalities involving the first Dirichlet eigenvalue, the perimeter, and the volume for different classes of sets. We give a complete description of the diagram for the class of open sets in Rd, basically showing that the isoperimetric and Faber-Krahn inequalities form a complete system of inequalities for these three quantities. We also give some qualitative results for the Blaschke-Santaló diagram for the class of planar convex domains: we prove that in this case the diagram can be described as the set of points contained between the graphs of two continuous and increasing functions. This shows in particular that the diagram is simply connected, and even horizontally and vertically convex. We also prove that the shapes that fill the upper part of the boundary of the diagram are smooth (C1,1), while those on the lower one are polygons (except for the ball). Finally, we perform some numerical simulations in order to have an idea on the shape of the diagram; we deduce from both theoretical and numerical results some new conjectures about geometrical inequalities involving the functionals under study in this paper.