The Daugavet and Delta-constants of points in Banach spaces

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.