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We investigate the geometry of the family $\cal M$ of isometry classes ofcompact metric spaces, endowed with the Gromov-Hausdorff metric. We show thatsufficiently small neighborhoods of generic finite spaces in the subspace ofall finite metric spaces with the same number of points are isometric to someneighborhoods in the space ${\mathbb R}^N_{\infty}$, i.e., in the space${\mathbb R}^N$ with the norm $\|(x_1,\ldots,x_N)\|=\max_i|x_i|$. As acorollary, we get that each finite metric space can be isometrically embeddedinto $\cal M$ in such a way that its image belongs to a subspace consisting ofall finite metric spaces with the same number $k$ of points. If the initialspace has $n$ points, then one can take $k$ as the least possible integer with$n\le k(k-1)/2$.

Local Structure of Gromov-Hausdorff Space, and Isometric Embeddings of Finite Metric Spaces into this Space