Minimizing curves in prox-regular subsets of Riemannian manifolds

TL;DR

This paper characterizes the proximal normal cone and tangent cones of prox-regular sets in Riemannian manifolds, and studies the properties of metric projections and minimizing curves within these sets.

Contribution

It provides new characterizations of normal and tangent cones, and establishes Lipschitz continuity and differentiability properties of metric projections in prox-regular sets.

Findings

Proximal normal cone characterized for prox-regular sets.

Metric projection is locally Lipschitz and directionally differentiable.

Necessary condition for minimizing curves in prox-regular sets derived.

Abstract

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 P_S 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 prox-regular set is derived.