  In this paper, we give a new characterization of the cut locus of a point on
a compact Riemannian manifold as the zero set of the optimal transport density
solution of the Monge-Kantorovich equations, a PDE formulation of the optimal
transport problem with cost equal to the geodesic distance. Combining this
result with an optimal transport numerical solver based on the so-called
dynamical Monge-Kantorovich approach, we propose a novel framework for the
numerical approximation of the cut locus of a point in a manifold. We show the
applicability of the proposed method on a few examples settled on 2d-surfaces
embedded in $R^{3}$ and discuss advantages and limitations.
