The Daugavet and Delta-constants of points in Banach spaces

TL;DR

This paper introduces the Daugavet and Δ-constants to quantify how points in Banach spaces relate to Daugavet and Δ-points, providing new insights into their geometric structure and examples in classical and Lipschitz-free spaces.

Contribution

It defines new localized constants for points in Banach spaces, linking them to global properties and providing explicit calculations and examples.

Findings

Existence of Banach spaces with all points having positive Daugavet constants despite zero Daugavet indices.

Exact values of Daugavet and Δ-constants in classical Banach spaces and Lipschitz-free spaces.

Identification of points with specific Daugavet and Δ-constants, including almost Daugavet and Δ-points.

Abstract

We introduce two new notions called the Daugavet constant and $\Delta$-constant of a point, which measure quantitatively how far the point is from being Daugavet point and $\Delta$-point and allow us to study Daugavet and $\Delta$-points in Banach spaces from a quantitative viewpoint. We show that these notions can be viewed as a localized version of certain global estimations of Daugavet and diametral local diameter two properties such as Daugavet indices of thickness. As an intriguing example, we present the existence of a Banach space $X$ in which all points on the unit sphere have positive Daugavet constants despite the Daugavet indices of thickness of $X$ being zero. Moreover, using the Daugavet and $\Delta$-constants of points in the unit sphere, we describe the existence of almost Daugavet and $\Delta$-points as well as the set of denting points of the unit ball. We also present exact values of the Daugavet and $\Delta$-constant on several classical Banach spaces, as well as Lipschitz-free spaces. In particular, it is shown that there is a Lipschitz-free space with a $\Delta$-point which is the furthest away from being a Daugavet point. Finally, we provide some related stability results concerning the Daugavet and $\Delta$-constant.