  In this paper we prove that every locally minimizing curve with constant
speed in a prox-regular subset of a Riemannian manifold is a weak geodesic.
Moreover, it is shown that under certain assumptions, every weak geodesic is
locally minimizing. Furthermore a notion of closed weak geodesics on
prox-regular sets is introduced and a characterization of these curves as
nonsmooth critical points of the energy functional is presented.
