Conformal dimension

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]

## Contents

* 1 Formal definition
* 2 Properties
* 3 Examples
* 4 See also
* 5 References

## Formal definition

Let X be a metric space and 𝒢 be the collection of all metric spaces that are quasisymmetric to X. The conformal dimension of X is defined as such

CdimX=infY∈𝒢dimHY

## Properties

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

dimTX≤CdimX≤dimHX

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 𝐑N 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

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

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