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Improved description of Blaschke--Santaló diagrams via numerical shape optimization

Ilias Ftouhi

Abstract

We propose a method based on the combination of theoretical results on Blaschke--Santaló diagrams and numerical shape optimization techniques to obtain improved description of Blaschke--Santaló 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 $λ_1$. 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 $(P,λ_1,A)$ and $(d,λ_1,A)$ where a strange phenomenon of non-continuity of the extremal shapes is observed.

Improved description of Blaschke--Santaló diagrams via numerical shape optimization

Abstract

We propose a method based on the combination of theoretical results on Blaschke--Santaló diagrams and numerical shape optimization techniques to obtain improved description of Blaschke--Santaló diagrams in the class of planar convex sets. To illustrate our approach, we study three relevant diagrams involving the perimeter , the diameter , the area and the first eigenvalue of the Laplace operator with Dirichlet boundary condition . The first diagram is a purely geometric one involving the triplet and the two other diagrams involve geometric and spectral functionals, namely and where a strange phenomenon of non-continuity of the extremal shapes is observed.
Paper Structure (32 sections, 14 theorems, 84 equations, 21 figures)

This paper contains 32 sections, 14 theorems, 84 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorems \ref{['th:main_1']} and \ref{['th:main_2']} and other theoretical results
  3. Some classical definitions and results of convex geometry
  4. The diagram D3
  5. A perturbation Lemma
  6. Elements of proof of Theorem \ref{['th:main_2']}
  7. A symmetry and regularity result
  8. Further results and comments
  9. The diagram D1
  10. Parametrizations of convex sets and numerical setting
  11. Parametrization via the support function
  12. Definitions and main properties
  13. Numerical setting
  14. Computation of the gradients
  15. Parametrization via the Gauge function
  16. ...and 17 more sections

Key Result

Theorem 2

We define The functions $\phi$ and $\psi$ are continuous and strictly increasing. Moreover,we have

Figures (21)

  • Figure 2: Perturbation field $V_{x_k}$ associated to the perturbation of the parameter $x_k$.
  • Figure 3: Convexity constraint via areas of the triangles.
  • Figure 4: Perturbation field $V_{\rho_k}$.
  • Figure 5: Approximation of the diagram $\mathcal{D}_1$ via random convex sets and some relevant shapes.
  • Figure 6: A symmetrical lens obtained as a solution of the problem $\max\{A(\Omega)\ |\ \Omega\in \mathcal{K},\ P(\Omega) = 2.4\ \text{and}\ d(\Omega) = 1\}$.
  • ...and 16 more figures

Theorems & Definitions (33)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Theorem 5: Steiner formula steiner
  • Theorem 6: Brunn--Minkowski inequality
  • Lemma 7
  • proof
  • Theorem 8
  • proof
  • ...and 23 more