Shortest path through random points

Let (M, g(1)) be a complete d-dimensional Riemannian manifold for d > 1. Let X-n be a set of n sample points in M drawn randomly from a smooth Lebesgue density f supported in M. Let x, y be two points in M. We prove that the normalized length of the power-weighted shortest path between x, y through X-n converges almost surely to a constant multiple of the Riemannian distance between x, y under the metric tensor g(p) = f(2)(1-P)/d g(1), where p > 1 is the power parameter.