Conformal dimension

In mathematics, the conformal dimension of a metric space X is the infimum of the Hausdorff dimension over the of X, that is, the class of all metric spaces quasisymmetric to X.^[1]

Formal definition[]

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

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

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 ${\displaystyle \mathbf {R} ^{N}}$ 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.

See also[]

• Anomalous scaling dimension

References[]