Shape optimization of polytopes and application to the polyhedral Saint–Venant inequality

Abstract We introduce a framework for shape optimization in the class of simple polytopes in arbitrary dimensions. We use a parametrization via hyperplanes and provide conditions for stability of simple polytopes under small perturbations of the hyperplanes. Next, we construct a mapping based on barycentric coordinates between the perturbed and reference polytopes. This allows us to use a transport theorem to compute the derivative of integrals defined on the polytope and perform the sensitivity analysis of shape functionals. This framework is applied to a numerical study of the polygonal and polyhedral Saint–Venant inequality. We solve an optimization problem that maximizes the torsion functional over polytopes with a fixed number of facets and equal volume. Using the previously defined parametrization via hyperplanes, the problem is reduced to a finite-dimensional optimization problem. A Proximal-Perturbed Lagrangian functional is employed to handle the volume constraint. In two dimensions, our numerical results support the conjecture stating that the solution is a regular polygon. In three dimensions, the numerical solutions converge towards either regular polyhedra or irregular polyhedra, depending on the number of faces.