Convexity of the Distance Function to Convex Subsets of Riemannian Manifolds

A characterization of the proximal normal cone is obtained and a separation theorem for convex subsets of Riemannian manifolds is established. Moreover, the convexity of the distance function $d_S$ for a convex subset S in the cases where the boundary of S contains a geodesic segment, the boundary of S is $C^2$ or the boundary of S is not regular is discussed. Furthermore, a nonsmooth version of positive semi-definiteness of the Hessian of convex functions on Riemannian manifolds is established.