The optimization of functionals depending on shapes which have convexity, diameter or constant width constraints poses difficulties from a numerical point of view. We show how to use the support function in order to approximate solutions to such problems by finite dimensional optimization problems under various constraints. After constructing the numerical framework, we present some applications from the field of convex geometry. We consider the optimization of various functionals depending on the volume, perimeter and Dirichlet Laplace eigenvalues under the constraints presented earlier. In particular we confirm numerically Meissner's conjecture, regarding three dimensional bodies of constant width with minimal volume, by directly solving an optimization problem.