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

TL;DR

This paper introduces a numerical shape optimization approach combined with theoretical insights to better describe Blaschke--Santaló diagrams for planar convex sets, revealing new extremal shape phenomena.

Contribution

It develops a novel method integrating theory and numerical optimization to improve the understanding of Blaschke--Santaló diagrams involving geometric and spectral functionals.

Findings

Enhanced descriptions of Blaschke--Santaló diagrams

Identification of non-continuity in extremal shapes

Application to diagrams involving perimeter, diameter, area, and eigenvalues

Abstract

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 $\lambda_1$. 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 $(P,\lambda_1,A)$ and $(d,\lambda_1,A)$ where a strange phenomenon of non-continuity of the extremal shapes is observed.