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

arXiv: https://arxiv.org/abs/2501.01228 · HTML: https://mlsanity.com/paper/2501.01228/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jan 2025.

## Built on by (most cited first, 2 of 2)

- [Numerical exploration of the range of shape functionals using neural networks](https://mlsanity.com/paper/2602.14881/) (2026)
- [Parametrizing Convex Sets Using Sublinear Neural Networks](https://mlsanity.com/paper/2605.03520/) (2026)

## Abstract (arXiv)

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.