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Local Structure of Gromov-Hausdorff Space, and Isometric Embeddings of Finite Metric Spaces into this Spaceстатья
- Авторы: Stavros Iliadis, Alexandr Ivanov, Alexey Tuzhilin
- Журнал: ArXiv e-prints
- Номер: 1604.07615
- Год издания: 2016
- Аннотация: We investigate the geometry of the family $\cM$ of isometry classes of compact metric spaces, endowed with the Gromov--Hausdorff metric. We show that sufficiently small neighborhoods of general position spaces, considered in the space of all finite metric spaces with the same number of points, are isometric to some neighborhoods of the space $\R^N_{\infty}$, i.e., of the space $\R^N$ with the norm $\|(x_1,\ldots,x_N)\|=\max_i|x_i|$. As a corollary, we get that each finite metric space can be isometrically embedded into $\cM$ in such a way that its image belongs to a subspace consisting of all finite metric spaces with the same number $k$ of points. If the initial space has $n$ points, then for the $k$ one can take the least possible integer number with $n\le k(k-1)/2$.
- Добавил в систему: Тужилин Алексей Августинович