Characterizations of Daugavet points and delta-points in Lipschitz-free spaces

Abstract

A norm 1 element x of a Banach space is a Daugavet point (respectively, a ∆-point) if every slice of the unit ball (respectively, every slice of the unit ball containing x) contains an element which is at distance almost 2 from x. We characterize Daugavet points and ∆-points in Lipschitz-free spaces. Furthermore, we construct a Lipschitz-free space with the Radon-Nikodým property and with a Daugavet point; this is the first known example of such a Banach space.

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

“…It is thus unclear whether a similar construction could be done on a weakly null normalized net, and in particular whether it could be implemented in $$\ell _1$$. Also, it is still unknown whether the Daugavet point in the space $$\mathcal {F}({\mathcal {M}})$$ from [41, Example 3.1] studied in Section 2 is a super $$\Delta$$‐point (see Question 7.1 in [32] for further discussions). So we do not know whether $$\ell _1$$ can be renormed with a super $$\Delta$$‐point. (b)If $$X$$ is a Banach space with a normalized weakly null Schauder basis $$(e_n)_{n\geqslant 1}$$, then the construction from Section 3 can be implemented in a natural way on this sequence in order to provide a renorming of $$X$$ for which $$e_1$$ is a super $$\Delta$$‐point and $$e_1^*$$ is a $${\rm weak}^*$$ super $$\Delta$$‐point.…”

“…Proof Fix $$\alpha \in (0,1)$$. By Lemma 6.6 and [41, Theorem 4.4], we have $$\left \Vert \mu \right \Vert _{\mathcal {F}({M,b_\alpha })} = 1-\alpha$$. Thus, since $$\mu$$ is a finitely supported element of $$\mathcal {F}({M,b_\alpha })$$, we may write $$\mu /\left \Vert \mu \right \Vert _{\mathcal {F}({M,b_\alpha })}$$ as a finite convex combination of $$b_\alpha$$‐molecules (e.g., by [42, Proposition 3.16]).…”

“…The relevance of the next lemma lies in the fact that condition (i) characterizes Δ-points when 𝜇 is a molecule (see [26,Theorem 4.7]) or, more generally, a finitely supported element of 𝑆  (𝑀) (see [41,Theorem 4.4]). Lemma 6.6.…”

“…In Section 5.1, we prove Theorem 5.6 and use it to improve results from [7] and [40] and show that asymptotically uniformly smooth spaces cannot contain $$\mathfrak {D}$$‐points and their duals cannot contain $${\rm weak}^*$$ $$\Delta$$‐points.…”

“…• 𝜇 can always be written as a convex sum of molecules (see, e.g., [42,Proposition 3.16]), and • 𝜇 also satisfies property (i) from Lemma 6.6, by [41,Theorem 4.4].…”