Improved description of Blaschke--Santal'o diagrams via numerical shape optimization

We propose a method based on the combination of theoretical results on Blaschke--Santal'o diagrams and numerical shape optimization techniques to obtain improved description of Blaschke--Santal'o diagrams in the class of planar convex sets. To illustrate our approach, we study three relevant diagrams involving the perimeter P, the diameter d, the area A and the first eigenvalue of the Laplace operator with Dirichlet boundary condition λ1. The first diagram is a purely geometric one involving the triplet (P,d,A) and the two other diagrams involve geometric and spectral functionals, namely (P,λ1,A) and (d,λ1,A) where a strange phenomenon of non-continuity of the extremal shapes is observed.