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A metric thickening of a given metric space X is any metric space admitting an isometric embedding of X. Thickenings have found use in applications of topology to data analysis, where one may approximate the shape of a dataset via the persistent homology of an increasing sequence of spaces.We introduce two new families of metric thickenings, the p-Vietoris-Rips and p-Čech metric thickenings for all 1 Ä p Ä 1, which include all probability measures on X whose p-diameter or p-radius is bounded from above, equipped with an optimal transport metric.The p-diameter (resp.p-radius) of a measure is a certain `p relaxation of the usual notion of diameter (resp.radius) of a subset of a metric space.These families recover the previously studied Vietoris-Rips and Čech metric thickenings when p D 1.As our main contribution, we prove a stability theorem for the persistent homology of p-Vietoris-Rips and p-Čech metric thickenings, which is novel even in the case p D 1.In the specific case p D 2, we prove a Hausmann-type theorem for thickenings of manifolds, and we derive the complete list of homotopy types of the 2-Vietoris-Rips thickenings of the n-sphere as the scale increases.55N31; 51F99, 53C23 1. Introduction 394 2. Background 398 3. The p-relaxation of metric thickenings 405 4. Basic properties 410 5. Stability 414 6.A Hausmann type theorem for 2-Vietoris-Rips and 2-Čech thickenings 423 7. The 2-Vietoris-Rips and 2-Čech thickenings of spheres with Euclidean metric 426 8. Bounding barcode length via spread 428 9. Conclusion
Adams et al. (Mon,) studied this question.