The radial behavior of a quasiconformal mapping (Q1331760)

The author proves three interesting theorems for quasiconformal maps in \(\mathbb{R}^n\), \(n\geq 2\). We state the first result here. Theorem: Let \(f\) be a \(K\)-quasiconformal mapping of \(B^n\) into \(\mathbb{R}^n\). Then for each \(0< \lambda\leq n- 1\) there exists a set of zero \(\lambda\)- dimensional Hausdorff measure on \(S^{n- 1}= \partial B^n\) such that the Hausdorff dimension of the image of \(S^{n- 1}\backslash E\) ist at most \(\lambda'\), where \(\lambda'< n\) depends only on the dilatation \(K\) of \(f\), \(\lambda\), and \(n\). In particular, there is a set \(F\subset S^{n- 1}\) of Hausdorff dimension zero such that the volume of \(f(S^{n -1}\backslash F)\) is zero.