Lipschitz slices versus linear slices in Banach spaces

Abstract

The aim of this note is to study the topology generated by Lipschitz slices in the unit sphere of a Banach space. We prove that the above topology agrees with the weak topology in the unit sphere and, as a consequence, we obtain easy Lipschitz characterizations of classical linear topics in Banach spaces as the Daugavet property, Radon-Nikodym property, convex point of continuity property and strong regularity, which shows that the above classical linear properties depend only on the natural uniformity in the Banach space given by the metric and the linear structure.