Delta‐points and their implications for the geometry of Banach spaces

Abstract

AbstractWe show that the Lipschitz‐free space with the Radon–Nikodým property and a Daugavet point recently constructed by Veeorg is in fact a dual space isomorphic to . Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of , with a ‐point. Building on these two results, we are able to renorm every infinite‐dimensional Banach space to have a ‐point. Next, we establish powerful relations between existence of ‐points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of ‐points for the asymptotic geometry of Banach spaces. In addition, we prove that if is a Banach space with a shrinking ‐unconditional basis with , or if is a Hahn–Banach smooth space with a dual satisfying the Kadets–Klee property, then and its dual fail to contain ‐points. In particular, we get that no Lipschitz‐free space with a Hahn–Banach smooth predual contains ‐points. Finally, we present a purely metric characterization of the molecules in Lipschitz‐free spaces that are ‐points, and we solve an open problem about representation of finitely supported ‐points in Lipschitz‐free spaces.