Fractal boundaries are not typical

Let M be a C1 n-dimensional compact submanifold of Rn. The boundary of M, ∂M, is itself a C1 compact (n−1)-dimensional submanifold of Rn. A carefully chosen set of deformations of ∂M defines a complete subspace consisting of boundaries of compact n-dimensional submanifolds of Rn, thus the Baire Category Theorem applies to the subspace. For the typical boundary element ∂W in this space, it is the case that ∂W is simultaneously nowhere-differentiable and of Hausdorff dimension n−1.