Bilipschitz mappings of nets.

Abstract

Let \(0<a<\sqrt 2\). Suppose \(\delta=\delta(d,\varepsilon)\) has the following property. If \(\mathcal N\) is an \(a\)-net of the Euclidean ball in \(\mathbb{R}^{d}\), \(A\subset \mathcal N\), and \(f:A\to \mathbb{R}^d\) is \((1+\varepsilon)\)-bilipschitz, then \(f\) admits a \((1+\delta)\)-bilipschitz extension \(f:\mathcal N\to \mathbb{R}^d\). We give some estimates of \(\delta\).