The regularity of the distance function propagates along minimizing geodesics

We consider the distance function from the boundary of an open bounded set Ω ⊂ R n associated to a Riemannian metric with C 1 , 1 coefficients. We show that the C 1 , 1 regularity propagates, towards the boundary ∂ Ω , along the distance minimizing geodesics. Hence, we show that the cut-locus is invariant with respect to the generalized gradient flow asso- ciated to the distance function and that it has the same homotopy type as Ω .