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A Complete Proof of the Limit Formula for Observable Diameter

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arxiv 2407.08122 v1 pith:JJNJJ5BZ submitted 2024-07-11 math.MG math.PR

Shigeaki Yokota This is my paper · ORCID
classification math.MGmath.PR
keywords formulalimitinequalityobservableproofcompleteconstructiveconvergence
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Ozawa and Shioya proposed the limit formula for observable diameters of pyramids under weak convergence. However, we find a constructive counterexample to an inequality used in their proof. In this paper, we correct the inequality and verify the limit formula.

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Cited by 2 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. 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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