Fractal boundaries are not typical

Let M be a C 1 n-dimensional compact submanifold of R n . The boundary of M, ∂M, is itself a C 1 compact ( n − 1 ) -dimensional submanifold of R n . A carefully chosen set of deformations of ∂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 ∂W in this space, it is the case that ∂W is simultaneously nowhere-differentiable and of Hausdorff dimension n − 1 .