Nonhomogeneous Euclidean first-passage percolation and distance learning

Consider an i.i.d. sample from unknown density function supported on manifold embedded in a high dimensional Euclidean space. We tackle the problem of learning distance between points, able to capture both geometry and underlying density. define such prove convergence, as size goes infinity, macroscopic one that we call Fermat it minimizes path functional, resembling principle optics. The proof boils down study geodesics first-passage percolation for nonhomogeneous Poisson point processes.