Triangulation and discretizations of metric measure spaces
We prove that a Ricci curvature based method of triangulation of compact Riemannian manifolds, due to Grove and Petersen, extends to the context of weighted Riemannian manifolds and more general metric measure spaces. In both cases the role of the lower bound on Ricci curvature is replaced by the curvature-dimension condition CD(K, N). Moreover, we show that the triangulation can be modified to become a thick one and that, in consequence, such manifolds admit weight-sensitive quasiregular mappings on Sn, with applications to information manifolds. The application of the existence of thick triangulation to estimating length and index of closed geodesics on the considered spaces is also explored.
Furthermore. we extend to weak CD(K, N) spaces the results of Kanai regarding the discretization of manifolds, and show that the volume growth of such a space is the same as that of any of its discretizations.