The Wall conjecture on domains in Euclidean spaces
The author proves the following property of a quasiconformal ball \(G\) in \(\mathbb{R}^n\), \(n\geq 2\): (*) \({\mathcal H}^{n-1} (\partial G \cap B(a,2r)) \geq r^{n-1}/c\) for each \(a\in G\) and \(r= d(a, \partial G)\) with \(c= c(n,K)\) where \(K\) is the coefficient of quasiconformality for \(G\) This was conjectured in [\textit{J. Heinonen}: Rev. Mat. Iberom. 12, No. 3, 697-725 (1996; Zbl 0872.30012)] and proved for \(n=3\); the case \(n=2\) is trivial. The author shows that the \((n-1)\)-dimensional Hausdorff measure \({\mathcal H}^{n-1}\) in (*) can be replaced by \({\mathcal H}_\infty^{n-1}\) which refers to the \((n-1)\)-dimensional Hausdorff content. The proof employs cubial homology. These results can be used to study the boundary absolute continuity of quasiconformal mappings.
scientific article; zbMATH DE number 1054319