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Quasisymmetric embeddings of products of cells into the Euclidean space. (English) Zbl 0807.30011

We shall mainly consider the following question. Suppose that \(A\) is a \(p\)-cell, \(B\) is a \(q\)-cell and there exists a quasisymmetric embedding \(f\colon A\times B\to \mathbb{R}^{p+q}\). What can we then say about the metric properties of \(A\) and \(B\)? It turns out that, for example, the cell \(A\) locally satisfies a \((p-1)\)-dimensional bounded turning condition. We also study the measures of \(A\) and \(B\) and show that if \(B\) is a quasiconvex arc, this implies certain conditions for the measure of \(A\).

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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