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

\( {\displaystyle \mathrm {Cdim} X=\inf _{Y\in {\mathcal {G}}}\dim _{H}Y} \)

Properties

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

\( {\displaystyle \dim _{T}X\leq \mathrm {Cdim} X\leq \dim _{H}X} \)

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.