Almost isometries of balls
Let \(\ell^n_2\) denote \(\mathbb{R}^n\) with the canonical Euclidean norm and \(B\) its unit ball. Let \(f\) be a bi-Lipschitz mapping of \(B\) into \(\ell^n_2\). The shape of \(f(B)\) in the case when both Lipschitz constants are close to one is investigated in the paper. The author also constructs an example of a mapping \(f\) with both Lipschitz constants arbitrarily close to one and such that \(f\) has distance (in the supremum norm) at least one from every isometry. This shows that a quasi-isometry can be far from all isometries.
scientific article; zbMATH DE number 1764705