# Delta- and Daugavet-points in Banach spaces

## TL;DR

This paper investigates the properties of Delta- and Daugavet-points in Banach spaces, establishing their equivalence in certain spaces, exploring their geometric implications, and introducing a new diameter two property with stability results.

## Contribution

It characterizes Delta- and Daugavet-points in various Banach spaces, introduces the convex diametral diameter two property, and studies its stability and implications.

## Key findings

- Delta- and Daugavet-points coincide in L1-spaces and M"untz spaces.
- Existence of Banach spaces where all unit sphere points are Delta-points but not Daugavet-points.
- The convex diametral diameter two property holds for C(K) spaces and M"untz spaces, and is stable under absolute sums.

## Abstract

A $\Delta$-point $x$ of a Banach space is a norm one element that is arbitrarily close to convex combinations of elements in the unit ball that are almost at distance $2$ from $x$. If, in addition, every point in the unit ball is arbitrarily close to such convex combinations, $x$ is a Daugavet-point. A Banach space $X$ has the Daugavet property if and only if every norm one element is a Daugavet-point. We show that $\Delta$- and Daugavet-points are the same in $L_1$-spaces, $L_1$-preduals, as well as in a big class of M\"untz spaces. We also provide an example of a Banach space where all points on the unit sphere are $\Delta$-points, but where none of them are Daugavet-points. We also study the property that the unit ball is the closed convex hull of its $\Delta$-points. This gives rise to a new diameter two property that we call the convex diametral diameter two property. We show that all $C(K)$ spaces, $K$ infinite compact Hausdorff, as well as all M\"untz spaces have this property. Moreover, we show that this property is stable under absolute sums.