The distance function in the presence of an obstacle

We study the Riemannian distance function from a fixed point (a point-wise target) of Euclidean space in the presence of a compact obstacle bounded by a smooth hypersurface. First, we show that such a function is locally semiconcave with a fractional modulus of order one half and that, near the obstacle, this regularity is optimal. Then, in the Euclidean setting, we show that the distance function is everywhere differentiable (except for the point-wise target) if and only if no obstacle is present. Finally, we prove that all the singular points of the distance function are not isolated, in the sense that each singularity belongs to a nontrivial continuum of singular points.