Nowhere differentiable Lipschitz maps and the Radon-Nikodým property (Q1336224)

The author proves that a Banach space \(X\) fails the RNP if and only if there is an equivalent norm \(||| \cdot |||)\) for \(X\) and an isometry \(U: \mathbb{R}\mapsto (X,||| \cdot |||\) such that \(U\) is nowhere differentiable on \(\mathbb{R}^ 1\). He also gives a characterization of closed convex sets which lack the RNP in terms of nowhere differentiability of Lipschitz maps.