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.

Let X be a metric space and ${\displaystyle {\mathcal {G}}}$ 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.

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