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On p-metric spaces and the p-Gromov-Hausdorff distance

2 Dec 2019arXiv:1912.00564links table onlyarchive 2025-07-28

Facundo Mémoli, Zhengchao Wan

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For each given p∈[1,∞] we investigate certain sub-family ℳₚ of the collection of all compact metric spaces ℳ which are characterized by the satisfaction of a strengthened form of the triangle inequality which encompasses, for example, the strong triangle inequality satisfied by ultrametric spaces. We identify a one parameter family of Gromov-Hausdorff like distances {d_(GH)⁽ᵖ⁾}_(p∈[1,∞]) on ℳₚ and study geometric and topological properties of these distances as well as the stability of certain canonical projections 𝔖ₚ:ℳ→ℳₚ. For the collection 𝒰 of all compact ultrametric spaces, which corresponds to the case p=∞ of the family ℳₚ, we explore a one parameter family of interleaving-type distances and reveal their relationship with {d_(GH)⁽ᵖ⁾}_(p∈[1,∞]).

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