Extremal structure and Duality of Lipschitz free spaces

TL;DR

This paper explores the extremal structure of Lipschitz free spaces, establishing new relationships between extremal points and their geometric properties, with implications for norm-attainment in vector-valued Lipschitz function spaces.

Contribution

It provides new insights into the extremal points of Lipschitz free spaces, including conditions under which molecules are extreme and the connection to denting points.

Findings

Every preserved extreme point is a denting point.

In certain cases, all extreme points are molecules.

Molecules are extreme when the segment [x, y] contains only x and y.

Abstract

We analyse the relationship between different extremal notions in Lipschitz free spaces (strongly exposed, exposed, preserved extreme and extreme points). We prove in particular that every preserved extreme point of the unit ball is also a denting point. We also show in some particular cases that every extreme point is a molecule, and that a molecule is extreme whenever the two points, say $x$ and $y$, which define it satisfy that the metric segment $[x, y]$ only contains $x$ and $y$. The most notable among them is the case when the free space admits an isometric predual with some additional properties. As an application, we get some new consequences about norm-attainment in spaces of vector valued Lipschitz functions.