Fractal boundaries are not typical

Let M be a C-1 n-dimensional compact submanifold of R-n. The boundary of M, partial derivative M, is itself a C-1 compact (n - 1)-dimensional submanifold of R-n. A carefully chosen set of deformations of partial derivative M defines a complete subspace consisting of boundaries of compact n-dimensional submanifolds of R-n, thus the Baire Category Theorem applies to the subspace. For the typical boundary element partial derivative W in this space, it is the case that partial derivative W is simultaneously nowhere-differentiable and of Hausdorff dimension n - 1. (C) 2006 Elsevier B.V. All rights reserved.