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.
Formal definition
Let X be a metric space and be the collection of all metric spaces that are quasisymmetric to X. The conformal dimension of X is defined as such
Properties
We have the following inequalities, for a metric space 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 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.