Numerical computation of the cut locus via a variational approximation of the distance function

We propose a new method for the numerical computation of the cut locus of a compact submanifold of ${\mathbb{R}}^3$ without boundary. This method is based on a convex variational problem with conic constraints, with proven convergence. We illustrate the versatility of our approach by the approximation of Voronoi cells on embedded surfaces of ${\mathbb{R}}^3$.