Typical Lipschitz images of rectifiable metric spaces.

This article studies typical 1-Lipschitz images of 푛-rectifiable metric spaces 퐸 into R m for m ≥ n . For example, if E ⊂ R k , we show that the Jacobian of such a typical 1-Lipschitz map equals 1 H n -almost everywhere and, if m > n , preserves the Hausdorff measure of 퐸. In general, we provide sufficient conditions, in terms of the tangent norms of 퐸, for when a typical 1-Lipschitz map preserves the Hausdorff measure of 퐸, up to some constant multiple. Almost optimal results for strongly 푛-rectifiable metric spaces are obtained. On the other hand, for any norm | ⋅ | on R m , we show that, in the space of 1-Lipschitz functions from ([ − 1 , 1 ] n , | ⋅ | ∞) to (R m , | ⋅ |) , the H n -measure of a typical image is not bounded below by any Δ > 0 . [ABSTRACT FROM AUTHOR]