SHORTEST PATH THROUGH RANDOM POINTS

Let (M, g1) be a complete d-dimensional Riemannian manifold for d > 1. Let Xn 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 Xn converges almost surely to a constant multiple of the Riemannian distance between x, y under the metric tensor gp = f2(1–p)/dg1, where p > 1 is the power parameter.