On the level sets of a distance function in a Minkowski space

Given a closed subset of an n-dimensional Minkowski space with a strictly convex or differentiable norm, then, for almost every r > 0, the r-level set (points whose distance from the closed set is r) contains an open subset which is an n I dimensional Lipschitz manifold and whose complement relative to the level set has n 1 dimensional Hausdorff measure zero. In case n = 2 and the norm is twice differentiable with bounded second derivative, almost every level set is a 1 manifold.