Annales Academiæ Scientiarum Fennicæ

Mathematica

Volumen 34, 2009, 319-345

# A SEWING PROBLEM IN METRIC SPACES

## Peter Haïssinsky

Université de Provence, LATP/CMI

39 rue Frédéric Joliot-Curie,
13453 Marseille Cedex 13, France;
phaissin 'at' cmi.univ-mrs.fr

**Abstract.**
This note is devoted to the solution of a sewing problem between metric spaces
sharing quasisymmetric copies of a given metric space. It is proved that the sewing yields a well-defined
conformal gauge, and we study properties inherited by the new space.
It follows from the construction that if *Y* is a closed uniformly perfect subset of
a proper metric space *X*, then, for any \epsilon > 0,
one can find a metric *d* in the conformal gauge of *X*
so that the Hausdorff dimensions of both (*X*,*d*) and
(*Y*,*d*) are \epsilon-close to their conformal
dimension.

**2000 Mathematics Subject Classification:**
Primary 30C65; Secondary 30C35, 54E40, 28A75.

**Key words:**
Sewing problem, quasisymmetric maps, conformal gauge.

**Reference to this article:** P. Haïssinsky:
A sewing problem in metric spaces.
Ann. Acad. Sci. Fenn. Math. 34 (2009), 319-345.

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