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Box distance and observable distance via optimal transport

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arxiv 2204.04893 v2 pith:AETTHHXU submitted 2022-04-11 math.MG

Hiroki Nakajima This is my paper · ORCID
classification math.MG
keywords metricmetricstransportdistanceobservableoptimaladditionexistence
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On the set of all metric measure spaces, we have two important metrics, the box metric and the observable metric, both introduced by M. Gromov. We obtain the representation of these metrics by using transport plan. In addition, we prove the existence of optimal transport plans of these metrics.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Radial Hyperbolic Measures: Shell Geometry, Pyramid Limits, and Gaussian Phase Transitions

    math.MG 2026-07 accept novelty 8.0 of 10

    A radial hyperbolic measure's pyramid limit is set by the effective radius s_n log(sinh ρ_n/√n), so intrinsic and wrapped hyperbolic Gaussians phase-transition at scales 1/n and 1/√n.

  2. Pyramids and Extended Metric Measure Spaces

    math.MG 2026-07 conditional novelty 8.0 of 10

    Every pyramid of metric measure spaces can be realized as the □-closure of the pyramid associated with an extended metric measure space, and concentrated pyramids have a unique representation.

  3. Poincar\'e Beta Balls: Radial Laws, Shell Transforms, and Phase Diagram

    math.MG 2026-07 accept novelty 7.0 of 10

    Rescaled Poincaré beta balls converge weakly to one of four pyramids—finite star trees, diameter ≤1, metric-transformed Gaussians, or the Gaussian pyramid—according to the A and βL balance.

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