The regularity of the distance function propagates along minimizing geodesics

We consider the distance function from the boundary of an open bounded set Ω⊂RnΩ⊂Rn associated to a Riemannian metric with C1,1C1,1 coefficients. We show that the C1,1C1,1 regularity propagates, towards the boundary ∂Ω∂Ω, along the distance minimizing geodesics. Hence, we show that the cut-locus is invariant with respect to the generalized gradient flow associated to the distance function and that it has the same homotopy type as ΩΩ.