Minimizing Curves in Prox-regular Subsets of Riemannian Manifolds

Abstract

We obtain a characterization of the proximal normal cone to a prox-regular subset of a Riemannian manifold and some properties of Bouligand tangent cones to these sets are presented. Moreover, we show that on an open neighborhood of a prox-regular set, the metric projection is locally Lipschitz and it is directionally differentiable at the boundary points of the set. Finally, a necessary condition for a curve to be a minimizing curve in a prox-regular set is derived.