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