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The optimization of shape functionals under convexity, diameter or constantwidth constraints shows numerical challenges. The support function can be usedin order to approximate solutions to such problems by finite dimensionaloptimization problems under various constraints. We propose a numericalframework in dimensions two and three and we present applications from thefield of convex geometry. We consider the optimization of functionals dependingon the volume, perimeter and Dirichlet Laplace eigenvalues under theaforementioned constraints. In particular we confirm numerically Meissner'sconjecture, regarding three dimensional bodies of constant width with minimalvolume.

Parametric Shape Optimization using the Support Function