  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.
