Sets with the Unique Footpoint Property and φ-Convex Subsets of Riemannian Manifolds

The concepts of φ-convexity and prox-regularity for sets are studied in the framework of finite-dimensional Riemannian manifolds. We prove that the two concepts coincide and these sets are contained in the class of sets with the unique footpoint property. Then by using a characterization of elements of proximal normal cone, we show that under certain conditions a set with the unique footpoint property is a φ-convex set.