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\\ l ess 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 als o 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 H ausdorff dimension exceeding one.