Before revision:   Finally, as another application, we connect this geometric persistent homology to the notion of filling radius of manifolds introduced by Gromov G07 and show some consequences related to (1) the homotopy type of the Vietoris-Rips complexes of spheres that follows from work of M.~Katz and (2) characterization (rigidity) results for spheres in terms of their Vietoris-Rips persistence diagrams .
After revision:  Finally, as another application, we connect this geometric persistent homology to the notion of filling radius of manifolds introduced by Gromov G07 and show some consequences related to (1) the homotopy type of the Vietoris-Rips complexes of spheres that follows from work of M.~Katz and (2) characterization (rigidity) results for spheres in terms of their Vietoris-Rips persistence barcodes which follow from work of F.~Wilhelm .