RADEMACHER'S THEOREM IN BANACH SPACES WITHOUT RNP

We improve a Duda's theorem concerning metric and w-Gateaux differentiability of Lipschitz mappings, by replacing the -ideal A of Aronszajn null sets [ARONSZAJN, N.: Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), 147{190], with the smaller -ideal ~ A of Preiss-Zajcek null sets [PREISS, D.|ZAJI CEK, L.: Directional derivatives of Lipschitz functions, Israel J. Math. 125 (2001), 1{27]. We also prove the inclusion C^{~o} \subset A^~, where C^{~o} is the \sigma-ideal of Preiss null sets [PREISS, D.: Gateaux differentiability of cone-monotone and pointwise Lipschitz functions, Israel J. Math. 203 (2014), 501{534].