Lipschitz images with fractal boundaries and their small surface wrapping

Abstract:

Assume $E\subset H\subset \mathbf {R}^{m}$ and $\Phi :E\to \mathbf {R}^{m}$ is a Lipschitz $L$-mapping; $|H|$ and $||H||$ denote the volume and the surface area of $H$. We verify that there exists a figure $F\supset \Phi (E)$ with $||F||\leq c_{L} ||H||$, and, of course, $|F|\leq c_{L} |H|$, where $c_{L}$ depends only on the dimension and on $L$. We also give an example when $E=H\subset \mathbf {R}^{2}$ is a square and $||\Phi (E)||=\infty$; in fact, the boundary of $\Phi (E)$ can contain a fractal of Hausdorff dimension exceeding one.