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 P , the diameter d , the area A and the first eigenvalue of the Laplace operator with Dirichlet boundary condition $$\lambda _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> . 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 $$(P,\lambda _1,A)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$(d,\lambda _1,A)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> where a strange phenomenon of non-continuity of the extremal shapes is observed.