## 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

### Keywords:

quasisymmetric map; bounded turning; quasiconvex arc
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