The following result can be proved as an application of the Kirchheim-Rademacher theorem along with the Kirchheim area formula.
Theorem. Suppose that $f:\mathbb{R}^n\supset\Omega\to X$, $\Omega$ open, is a Lipschitz continuous map onto a metric space $X$, $f(\Omega)=X$. Then $\operatorname{dim} X=n$ if and only if $\mathcal{H}^n(X)>0$.
For further comments, see Topological dimension, Hausdorff dimension, and Lipschitz mappings.
- Length preserving mappings