# Conformal dimension

From HandWiki

In mathematics, the **conformal dimension** of a metric space *X* is the infimum of the Hausdorff dimension over the conformal gauge of *X*, that is, the class of all metric spaces quasisymmetric to *X*.^{[1]}

## Formal definition

Let *X* be a metric space and [math]\displaystyle{ \mathcal{G} }[/math] be the collection of all metric spaces that are quasisymmetric to *X*. The conformal dimension of *X* is defined as such

- [math]\displaystyle{ \mathrm{Cdim} X = \inf_{Y \in \mathcal{G}} \dim_H Y }[/math]

## Properties

We have the following inequalities, for a metric space *X*:

- [math]\displaystyle{ \dim_T X \leq \mathrm{Cdim} X \leq \dim_H X }[/math]

The second inequality is true by definition. The first one is deduced from the fact that the topological dimension T is invariant by homeomorphism, and thus can be defined as the infimum of the Hausdorff dimension over all spaces homeomorphic to *X*.

## Examples

- The conformal dimension of [math]\displaystyle{ \mathbf{R}^N }[/math] is
*N*, since the topological and Hausdorff dimensions of Euclidean spaces agree. - The Cantor set
*K*is of null conformal dimension. However, there is no metric space quasisymmetric to*K*with a 0 Hausdorff dimension.

## See also

- Anomalous scaling dimension

## References

- ↑ John M. Mackay, Jeremy T. Tyson,
*Conformal Dimension : Theory and Application*, University Lecture Series, Vol. 54, 2010, Rhodes Island

Original source: https://en.wikipedia.org/wiki/Conformal dimension.
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