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Porous sets and quasisymmetric maps. (English) Zbl 0617.30025

A set A in \({\mathbb{R}}^ n\) is called to be \(\alpha\)-porous \((0<\alpha \leq 1)\), if each closed ball \(\bar B(x,r)\) in \({\mathbb{R}}^ n\) contains a point z such that the open ball \(B(z,\alpha r)\) does not meet A. For example, \({\mathbb{R}}^ p\) is 1-porous in \({\mathbb{R}}^{p+1}\), while the Cantor middle- third set is (1/4)-porous in \({\mathbb{R}}^ 1.\)
One of the main results in this paper is the following: Let the set A be \(\alpha\)-porous as above. If a mapping \(f: A\to {\mathbb{R}}^ n\) is \(\eta\)- symmetric, then the image fA is \(\alpha_ 1\)-porous, where \(\alpha_ 1\) is a positive constant depending only on \(\alpha\) and \(\eta\), but not on n. In particular, every Lipschitzian map preserves the porosity of sets.
Reviewer: K.Shibata

MSC:

30C62 Quasiconformal mappings in the complex plane
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
31B99 Higher-dimensional potential theory
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