The Wall conjecture on domains in Euclidean spaces
type: Item
instance of: scholarly article
title: The Wall conjecture on domains in Euclidean spaces
zbMATH Open document ID: 0878.30013
DOI: 10.1007/BF02677489
published in: Manuscripta Mathematica
publication date: 1998-01-11T00:00:00Z
full work available at URL: https://eudml.org/doc/156317
review text: 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.
Mathematics Subject Classification ID: 30C65
zbMATH DE Number: 1054319
author: Jussi Väisälä
reviewed by: O. Martio
Wikidata QID: Q123160114
MaRDI profile type: Publication
cites work: Q4302436
cites work: Q5565773
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OpenAlex ID: W2050485911
description: scientific article; zbMATH DE number 1054319