On ℓ_p-Vietoris-Rips complexes and blurred magnitude homology

One of the main tools in topological data analysis is the notion of a persistence module. The most prominent example is the persistence module associated with the Vietoris–Rips complex of a finite metric space. On the other hand, the concept of magnitude has become increasingly well known in data analysis. Recently, Nina Otter introduced blurred magnitude homology, which is also a persistence module associated with a metric space. Govc and Hepworth showed that the magnitude of a finite metric space can be uniquely recovered from its blurred magnitude homology. For 1 ≤ p ≤ ∞, we define the ℓ_p-Vietoris–Rips complexes and the associated ℓ_p​-persistent homology of metric spaces, and we study their fundamental properties. We show that for p=∞ this theory recovers the classical theory of Vietoris–Rips complexes and their persistent homology, while for p=1 it recovers the theory of blurred magnitude homology.