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:  Our results also permit proving several bounds on the length of intervals in the Vietoris-Rips barcode by other metric invariants. 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 .