Lipschitz images with fractal boundaries and their small surface wrapping

Assume E subset of H subset of R(m) and Phi : E --> 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 superset of Phi(E) with \\F\\ less than or equal to c(L)\\H\\, and, of course, \F\ less than or equal to c(L)\H\, where c(L) depends only on the dimension and on L. We also give an example when E = H subset of R(2) is a square and \\Phi(E)\\ = infinity; in fact, the boundary of Phi(E) can contain a fractal of Hausdorff dimension exceeding one.