This article studies typical 1-Lipschitz images of $n$-rectifiable metric spaces $E$ into $\mathbb{ R}^m$ for $m\geq n$. For example, if $E\subset \mathbb{ R}^k$, we show that the Jacobian of such a typical 1-Lipschitz map equals 1 $\mathcal{ H}^n$-almost everywhere and, if $m>n$, preserves the Hausdorff measure of $E$. In general, we provide sufficient conditions, in terms of the tangent norms of $E$, for when a typical 1-Lipschitz map preserves the Hausdorff measure of $E$, up to some constant multiple. Almost optimal results for strongly $n$-rectifiable metric spaces are obtained. On the other hand, for any norm $|\cdot|$ on $\mathbb{ R}^m$, we show that, in the space of 1-Lipschitz functions from $([-1,1]^n,|\cdot|_\infty)$ to $(\mathbb{ R}^m,|\cdot|)$, the $\mathcal{ H}^n$-measure of a typical image is not bounded below by any $\Delta>0$.
Categories: math.MG math.FA