Nowhere Differentiable Lipschitz Maps and the Radon-Nikodým Property

We prove that a Banach space X fails the RNP if and only if there is an equivalent norm |||•||| for X and an isometry U: R1 ↦ (X, |||•|||) such that U is nowhere differentiable on R1. A characterization of closed convex sets which lack the RNP is also given in terms of nowhere differentiability of Lipschitz maps.