In the applied algebraic topology community, the persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces. In this paper, we consider a different, more geometric way of generating persistent homology of metric spaces which arises by first embedding a given metric space into a larger space and then considering thickenings of the original space. In the course of doing this, we construct an appropriate category for studying this notion of persistent homology and show that, in a category theoretic sense, the standard persistent homology of the Vietoris-Rips filtration is isomorphic to our geometric persistent homology provided that the embedding space satisfies a property called hyperconvexity. As an application of this isomorphism we are able to give succint proofs of the characterization of the persistent homology of products and joins of metric spaces. Finally, as another application, we connect this geometric persistent homology to the notion of filling radius of manifolds introduced by Gromov [26] 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.

We provide both the precise value and general upper and lower bounds for the Gromov-Hausdorff distance d GH pS m , S n q between spheres S m and S n (endowed with the round metric) for 0 ď m ă n ď 8. Some of these lower bounds are based on certain topological ideas related to the Borsuk-Ulam theorem. Via explicit constructions of (optimal) correspondences we prove that our lower bounds are tight in the cases of d GH pS 0 , S n q, d GH pS m , S 8 q, d GH pS 1 , S 2 q, d GH pS 1 , S 3 q and d GH pS 2 , S 3 q. We also formulate a number of open questions. CONTENTS 1. Introduction 1.1. Overview of our results 1.2. Discussion 1.3. Acknowledgements 2. Preliminaries 2.1. Notation and conventions about spheres. 3. Some general lower bounds 3.1. The proof of Proposition 1.3 3.2. Other lower bounds 4. The proof of Theorem 1 Space filling curves. Spherical suspensions. The proof of Theorem 6. 5. A Borsuk-Ulam theorem for discontinuous functions and the proof of Theorem 2 5.1. A succinct proof of Theorem 8 5.2. The proofs of Theorem 2 and 3 6.

We study a generalization of the classical Multidimensional Scaling procedure to the setting of general metric measure spaces. We identify spectral properties of the generalized cMDS operator thus providing a natural and rigorous mathematical formulation of cMDS. Furthermore, we characterize the cMDS output of several continuous exemplar metric measures spaces. In particular, we characterize the cMDS output for spheres S d−1 (with geodesic distance) and subsets of Euclidean space. In particular, the case of spheres requires that we establish the its cMDS operator is trace class, a condition which is natural in context when the cMDS has infinite rank (such as in the case of spheres with geodesic distance). Finally, we establish the stability of the generalized cMDS process with respect to the Gromov-Wasserstein distance.

The smallest hyperconvex metric space containing a given metric space X is called the tight span of X. It is known that tight spans have many nice geometric and topological properties, and they are gradually becoming a target of research of both the metric geometry community and the topological/geometric data analysis community. In this paper, we study the tight span of n-spheres (with either geodesic metric or ∞ -metric).