Minimizing Curves in Prox-regular Subsets of Riemannian Manifolds

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

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