Estimating the volume of a Riemann manifold from sampled distances

Question

Suppose we are given the pairwise distance matrix of $N$ points that have been uniformly sampled from an unknown Riemann manifold $X$, where the dimension $D$ of the manifold is not known. Can we estimate the volume of $X$? I suppose we could first estimate $D$ and then sum the volume of $D$-dimensional balls centered on each point with diameter equal to the distance to the nearest neighbor. But are there more direct approaches?