A pathological example of a uniform quotient mapping between Euclidean spaces (Q5939282)

Let \(X\), \(Y\) be metric spaces. The ball of radius \(r\) centered at \(x\) is denoted by \(B_r(x)\). A map \(f: X\to Y\) is called co-uniform if there is a positive function \(\omega\) such that \[ f(B_r(x))\supset B_{\omega(r)}(f(x)) \] for every \(x\in X\) and every positive \(r\). If there is a positive constant \(c\) such that \[ B_{cr}(f(x))\supset f(B_r(x)) \] for every \(x\in X\) and every positive \(r\) then \(f\) is called Lipschitz. The author proves the following theorem, which extends a result by \textit{S. Bates}, \textit{W. B. Johnson}, \textit{J. Lindenstrauss}, \textit{D. Preiss} and \textit{G. Schechtman} [Geom. Funct. Anal. 9, No. 6, 1092-1127 (1999; Zbl 0954.46014)], saying that there exists a Lipschitz and co-uniform mapping from \(\mathbb{R}^3\) onto \(\mathbb{R}^2\), which annihilates the unit ball of a hyperplane. Theorem. Let \(n\geq 1\). There exists a Lipschitz mapping \(T\) from \(\mathbb{R}^{n+2}= \mathbb{R}^{n+1}\oplus \mathbb{R}\) onto \(\mathbb{R}^{n+1}\) such that \(T\) is a co-uniform and \(T(B^E_1(0))= \{0\}\), where \(E= \mathbb{R}^{n+1}\oplus 0\). Here all \(\mathbb{R}^m\) are endowed with the canonical Euclidean norm.