Delicate symmetry (Q750905)

Let d,a,b \(>0\), \(S\subset {\mathbb{R}}^ 2\) such that every open disc of diameter d contains at least one element of S and f: \(S\to {\mathbb{R}}^ 2\) a map such that \(a| x-y| \leq | f(x)-f(y)| \leq b| x- y|\) for all \(x,y\in {\mathbb{R}}^ 2\). There is proved that then every open disc of diameter \(20b^ 2d/a\) contains at least one point belonging to f(S).