Improved description of Blaschke--Santaló diagrams via numerical shape optimization
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915088050421760 |
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| author | Ftouhi, Ilias |
| author_facet | Ftouhi, Ilias |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_01228 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved description of Blaschke--Santaló diagrams via numerical shape optimization Ftouhi, Ilias Optimization and Control 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. |
| title | Improved description of Blaschke--Santaló diagrams via numerical shape optimization |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2501.01228 |