  We characterize the differentiable points of the distance function from a
closed subset $N$ of an arbitrary dimensional Finsler manifold in terms of the
number of $N$-segments. In the case of a 2-dimensional Finsler manifold, we
prove the structure theorem of the cut locus of a closed subset $N$, namely
that it is a local tree, it is made of countably many rectifiable Jordan arcs
except for the endpoints of the cut locus and that an intrinsic metric can be
introduced in the cut locus and its intrinsic and induced topologies coincide.
  We should point out that these are new results even for Riemannian manifolds.
