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".
Formal definition.
Let "X" be a metric space and formula_1 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".