# Delta-points and their implications for the geometry of Banach spaces

## TL;DR

This paper investigates the structure of Banach spaces with $ ext{Delta}$-points, establishing their existence, dual relations, and geometric implications, including characterizations within Lipschitz-free spaces and renormings of classical spaces.

## Contribution

It demonstrates the existence of $ ext{Delta}$-points in superreflexive spaces, characterizes $ ext{Delta}$-points in Lipschitz-free spaces, and explores their impact on Banach space geometry.

## Key findings

- Constructed a dual space with a $ ext{Delta}$-point isomorphic to $ ext{ell}_1$.
- Proved the existence of $ ext{Delta}$-points in renormed $ ext{ell}_2$ spaces.
- Established conditions under which Banach spaces and their duals lack $ ext{Delta}$-points.

## Abstract

We show that the Lipschitz-free space with the Radon--Nikod\'{y}m property and a Daugavet point recently constructed by Veeorg is in fact a dual space isomorphic to $\ell_1$. Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of $\ell_2$, with a $\Delta$-point. Building on these two results, we are able to renorm every infinite-dimensional Banach space with a $\Delta$-point. Next, we establish powerful relations between existence of $\Delta$-points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of $\Delta$-points for the asymptotic geometry of Banach spaces. In addition, we prove that if $X$ is a Banach space with a shrinking $k$-unconditional basis with $k < 2$, or if $X$ is a Hahn--Banach smooth space with a dual satisfying the Kadets--Klee property, then $X$ and its dual $X^*$ fail to contain $\Delta$-points. In particular, we get that no Lipschitz-free space with a Hahn--Banach smooth predual contains $\Delta$-points. Finally we present a purely metric characterization of the molecules in Lipschitz-free spaces that are $\Delta$-points, and we solve an open problem about representation of finitely supported $\Delta$-points in Lipschitz-free spaces.