| Summary: | It is known that the $W^{1,p}$-distance between an orientation-preserving mapping in $W^{1,p}(\Omega ;\mathbb{R}^n)$ and another orientation-preserving mapping $\Theta \in C^1(\overline{\Omega };\mathbb{R}^n)$, where $\Omega $ is a domain in $\mathbb{R}^n$, $n\geqslant 2$, and $p>1$ is a real number, is bounded above by the $L^p$-distance between the square roots of the metric tensor fields induced by these mappings, multiplied by a constant depending only on $p$, $\Omega $, and $\Theta $.The object of this Note is to establish a better inequality of this type, and to provide in addition an explicitly computable upper bound on the constant appearing in it. An essential role is played in our proofs by the notion of geodesic distance inside an open subset of $\mathbb{R}^n$.
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