Local geometry of the Gromov-Hausdorff metric space and totally asymmetric finite metric spaces (Q2055333)

This paper studies the Gromov-Hausdorff space \(\mathcal M\). A finite totally asymmetric metric space \(M=\{1,\ldots,n\}\) represents a point in \(\mathcal M\). The author shows that a neighborhood of \(M\) in \(\mathcal M\) among spaces with \(n\) points is isometric to a neighborhood in \(\mathbb R^{n(n-1)/2}\) equipped with the \(\ell^{\infty}\) norm. This is used to show that every finite metric space \(X\) embeds in a neighborhood of a finite totally asymmetric metric space in \(\mathcal M\).