REGULARITY OF GEODESICS IN SETS OF POSITIVE

The question of regularity of geodesics is a central theme in geometry, having been addressed fully in the context of smooth Riemannian manifolds with and without boundary. In this paper, we examine geodesics in regular sets of positive reach (PR sets) in Euclidean space, and we prove that any geodesic in such a set has Lipschitz continuous velocity. To prove the result, we first show that the velocity of a geodesic has essentially bounded variation; we then use the geometric structure of the associated tangent and normal cones to conclude that the first derivative is Lipschitz continuous.