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 Fundam. Prikl. Mat., 2019, Volume 22, Issue 6, Pages 263–272 (Mi fpm1863)

Local geometry of the Gromov–Hausdorff metric space and totally asymmetric finite metric spaces

A. M. Filin

Lomonosov Moscow State University

Abstract: In the present paper, we investigate the structure of the metric space $\mathcal M$ of compact metric spaces considered up to an isometry and endowed with the Gromov–Hausdorff metric in a neighbourhood of a finite metric space, whose isometry group is trivial. It is shown that a sufficiently small ball in the subspace of $\mathcal M$ consisting of finite spaces with the same number of points centered at such a space is isometric to a corresponding ball in the space $\mathbb R^N$ endowed with the norm $|(x_1, …, x_N ) | = \max\limits_{i} |x_i|$. Also an isometric embedding of a finite metric space into a neighbourhood of a finite asymmetric space in $\mathcal M$ is constructed.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 075-02-2018-867 Russian Foundation for Basic Research 16-01-00378_a

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UDC: 514.13+519.173

Citation: A. M. Filin, “Local geometry of the Gromov–Hausdorff metric space and totally asymmetric finite metric spaces”, Fundam. Prikl. Mat., 22:6 (2019), 263–272

Citation in format AMSBIB
\Bibitem{Fil19} \by A.~M.~Filin \paper Local geometry of the Gromov--Hausdorff metric space and totally asymmetric finite metric spaces \jour Fundam. Prikl. Mat. \yr 2019 \vol 22 \issue 6 \pages 263--272 \mathnet{http://mi.mathnet.ru/fpm1863}