Almost Isometries of Balls

Let f be a bi-Lipschitz mapping of the Euclidean ball BRn into ℓ2 with both Lipschitz constants close to one. We investigate the shape of f(BRn). We give examples of such a mapping f, which has the Lipschitz constants arbitrarily close to one and at the same time has in the supremum norm the distance at least one from every isometry of Rn.