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Gromov-Hausdorff distances between normed spaces

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arxiv 2407.01388 v1 pith:RXYFHYUN submitted 2024-07-01 math.MG

I. N. Mikhailov This is my paper · ORCID
classification math.MG
keywords normedspacesgromov-hausdorffdistancefinite-dimensionalfiniteotherprove
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In the present paper we study the original Gromov-Hausdorff distance between real normed spaces. In the first part of the paper we prove that two finite-dimensional real normed spaces on a finite Gromov-Hausdorff distance are isometric to each other. We then study the properties of finite point sets in finite-dimensional normed spaces whose cardinalities exceed the equilateral dimension of an ambient space. By means of the obtained results we prove the following enhancement of the aforementioned theorem: every finite-dimensional normed space lies on an infinite Gromov-Hausdorff distance from all other non-isometric normed spaces.

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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. When the Gromov-Hausdorff distance between finite-dimensional space and its subset is finite?

    math.MG 2024-11 conditional novelty 6.0 of 10

    A subset A of finite-dimensional Euclidean space has finite Gromov-Hausdorff distance to the whole space if and only if A is an epsilon-net for some epsilon.

  2. Ultrametric spaces and clouds

    math.MG 2025-01 conditional novelty 5.0 of 10

    The ultrametrization map U is 1-Lipschitz on all metric spaces, preserves products with dotted connected spaces, and forces mutual exclusion of ultrametric and dotted connected spaces in unbounded clouds.

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