In this paper, we propose a method based on the combination of theoretical results on Blaschke-Santaló diagrams and numerical shape optimization techniques to obtain optimal description of Blaschke-Santaló diagrams in the class of convex sets. To illustrate our approach, we study three relevant diagrams involving the perimeter P , the diameter d, the volume | • | 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, | • |) and the two other diagrams involve geometric and spectral functionals, namely (P, λ1, | • |) (studied in [1, 24]) and (d, λ1, | • |) where a strange phenomenon of non-continuity of the extremal shapes is observed.