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]

Formal definition

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

$\mathrm{Cdim} X = \inf_{Y \in \mathcal{G}} \dim_H Y$

Properties

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

$\dim_T X \leq \mathrm{Cdim} X \leq \dim_H X$

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 $\mathbf{R}^N$ 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.