On the numerical approximation of Blaschke-Santaló diagrams using Centroidal Voronoi Tessellations

Blaschke-Santaló diagrams are images of maps defined on a set of parameters, taking values into an Euclidean space. Typically, the dimension of the source space is high, possibly infinite, while the target space is two or three dimensional. These diagrams help characterize geometrically various inequalities and are of particular interest in the field of shape optimization. We propose a numerical method, based on Centroidal Voronoi Tessellations, which produces sample points in the parameter space that have uniformly distributed images in the Blaschle-Santaló diagram, therefore providing an accurate description of the latter. Compared with the classical Monte Carlo methods, which simply use a large number of images corresponding to random parameters, the method proposed is computationally efficient and precise. Simulations for two and three dimensional diagrams are presented involving examples in algebra and shape optimization.