Lipschitz slices versus linear slices in Banach spaces

TL;DR

This paper explores the topology generated by Lipschitz slices in Banach spaces, showing it coincides with the weak topology and providing Lipschitz characterizations of key linear properties.

Contribution

It demonstrates that classical linear properties in Banach spaces can be characterized using Lipschitz slices, linking nonlinear and linear topologies.

Findings

Lipschitz slices generate the same topology as the weak topology in the unit sphere.

Lipschitz characterizations of Radon-Nikodym property, convex point of continuity, and strong regularity.

Classical linear properties depend only on the metric and linear structure, not on linearity itself.

Abstract

The aim of this note is 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 Lipschitz characterizations of classical linear topics in Banach spaces, as Radon-Nikodym property, convex point of continuity property and strong regularity, which shows that the above classical linear properties only depend on the natural uniformity in the Banach space given by the metric and the linear structure.