The lecture notes of J. Harrison claim to generalize certain theorems on smooth domains, such as Stokes' theorem, to non-smooth (even fractal) domains. The theory is built on objects called polyhedral k-chains or chainlets, which are formal sums of convex hulls spanned by k+1 vertices in $\mathbb{R}^n$.
But how can one describe arbitrary sets, or manifolds, using k-chainlets? E.g. how is it possible to write down an expression for a sphere (which is completely smooth and has no vertices) using (perhaps infinite) sums of k-cells/chains (which have, by definition, tons of vertices)?
While Whitney's decomposition allows for the decomposition of open sets into smaller cubes (which are k-cells), decomposition of closed sets into polyhedral k-chains or chainlets is unclear to me. Therefore, the question remains: Is there a decomposition or approximation theorem available for arbitrary (closed, nowhere differentiable, smooth etc.) sets?