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Parametric shape optimization using the support function

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Abstract

The optimization of shape functionals under convexity, diameter or constant width constraints shows numerical challenges. The support function can be used in order to approximate solutions to such problems by finite dimensional optimization problems under various constraints. We propose a numerical framework in dimensions two and three and we present applications from the field of convex geometry. We consider the optimization of functionals depending on the volume, perimeter and Dirichlet Laplace eigenvalues under the aforementioned constraints. In particular we confirm numerically Meissner’s conjecture, regarding three dimensional bodies of constant width with minimal volume.

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Authors and Affiliations

  1. Secção de Matemática, Departmento de Ciências e Tecnologia, Universidade Aberta, Rua da Escola Politécnica 141-147, 1269-001, Lisbon, Portugal

    Pedro R. S. Antunes

  2. Group of Mathematical Physics, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Edifício C6, 1749-016, Lisbon, Portugal

    Pedro R. S. Antunes

  3. Centre de Mathématiques Appliquées, École Polytechnique, Rue de Saclay, 91128, Palaiseau, France

    Beniamin Bogosel

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  1. Pedro R. S. Antunes
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  2. Beniamin Bogosel
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Correspondence to Beniamin Bogosel.

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Antunes, P.R.S., Bogosel, B. Parametric shape optimization using the support function. Comput Optim Appl 82, 107–138 (2022). https://doi.org/10.1007/s10589-022-00360-4

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