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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10250_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>λ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10250_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$(P,\lambda _1,A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10250_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$(d,\lambda _1,A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> where a strange phenomenon of non-continuity of the extremal shapes is observed.

Improved Description of Blaschke–Santaló Diagrams via Numerical Shape Optimization

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2025年4月2日

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Ilias Ftouhi

摘要

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 $\lambda _1$ $λ_{1}$ . 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 $(P,\lambda _1,A)$ $(P, λ_{1}, A)$ and $(d,\lambda _1,A)$ $(d, λ_{1}, A)$ where a strange phenomenon of non-continuity of the extremal shapes is observed.