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Quasi-Symmetric Embeddings in Euclidean Spaces

Jussi Väisälä
Transactions of the American Mathematical Society
Vol. 264, No. 1 (Mar., 1981), pp. 191-204
DOI: 10.2307/1998419
Stable URL: http://www.jstor.org/stable/1998419
Page Count: 14
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Quasi-Symmetric Embeddings in Euclidean Spaces
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Abstract

We consider quasi-symmetric embeddings $f: G \rightarrow R^n, G$ open in $R^p, p \leqslant n$. If $p = n$, quasi-symmetry implies quasi- conformality. The converse is true if $G$ has a sufficiently smooth boundary. If $p < n$, the Hausdorff dimension of $fG$ is less than $n$. If $fG$ has a finite $p$-measure, $f$ preserves the property of being of $p$- measure zero. If $p < n$ and $n \geqslant 3, R^n$ contains a quasi-symmetric $p$-cell which is topologically wild. We also prove auxiliary results on the relations between Hausdorff measure and Cech cohomology.

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