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Piotr Hajlasz
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I will mention sixseven different applications:

I will mention six different applications:

I will mention seven different applications:

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Martin Sleziak
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[AA] L. Ambrosio, G. Alberti, A geometricgeometrical approach to monotone functionfunctions in $\mathbb{R}^n$. Math Z. 230(1999), 259-316, DOI: 10.1007/PL00004691.

[AA] L. Ambrosio, G. Alberti, A geometric approach to monotone function in $\mathbb{R}^n$. Math Z. 230(1999), 259-316.

[AA] L. Ambrosio, G. Alberti, A geometrical approach to monotone functions in $\mathbb{R}^n$. Math Z. 230(1999), 259-316, DOI: 10.1007/PL00004691.

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Piotr Hajlasz
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  • Topological dimension

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
  • Length preserving mappings
  • Topological dimension

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
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