Conformal dimension: Difference between revisions

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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 $X$ .^[1]

Formal definition

Let $X$ be a metric space and ${\mathcal {G}}$ be the collection of all metric spaces that are quasisymmetric to $X$ . The conformal dimension of $X$ is defined as such

$CdimX=inf_{Y\in {\mathcal {G}}}dim_{H}Y$

Properties

We have the following inequalities, for a metric space $X$ :

$dim_{T}X\leq CdimX\leq dim_{H}X$

Examples

References

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

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

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-The '''conformal dimension''' of a metric space <math>X</math> is the infimum of the Hausdorff dimension over the conformal gauge of <math>X</math>, that is, the class of all metric spaces quasisymmetric to <math>X</math>.<ref>John M. Mackay, Jeremy T. Tyson, ''Conformal Dimension : Theory and Application'', University Lecture Series, Vol. 54, 2010, Rhodes Island</ref>
+The '''conformal dimension''' of a metric space <math>X</math> is the infimum of the [[Hausdorff dimension]] over the conformal gauge of <math>X</math>, that is, the class of all metric spaces quasisymmetric to <math>X</math>.<ref>John M. Mackay, Jeremy T. Tyson, ''Conformal Dimension : Theory and Application'', University Lecture Series, Vol. 54, 2010, Rhodes Island</ref>
+==Formal definition==
+Let <math>X</math> be a metric space and <math>\mathcal{G}</math> be the collection of all metric spaces that are quasisymmetric to <math>X</math>. The conformal dimension of <math>X</math> is defined as such
+<math>Cdim X = inf_{Y \in \mathcal{G}} dim_H Y</math>
+==Properties==
+We have the following inequalities, for a metric space <math>X</math> :
+<math>dim_T X \leq Cdim X \leq dim_H X</math>
+==Examples==
 ==See also==