Before revision:  New results include a characterization of convergence with respect to transport metrics, the existence of Fr\'echet means  for any distribution of diagrams, and an exhaustive description of continuous linear representations of persistence diagrams.
After revision:  New results include a characterization of convergence with respect to Wasserstein metrics, a geometric description of barycenters ( Fr\'echet means for any distribution of diagrams, and an exhaustive description of continuous linear representations of persistence diagrams.