[Paper Review] Daugavet- and Delta-points in spaces of Lipschitz functions

Triinu Veeorg|arXiv (Cornell University)|Jun 7, 2022

Advanced Banach Space Theory4 citations

TL;DR

This paper establishes the equivalence of Daugavet- and Δ-points in spaces of Lipschitz functions over proper metric spaces, providing characterizations for these points and proving that Δ-points exist in all infinite-dimensional Lipschitz function spaces, while Daugavet-points do not necessarily exist. It further demonstrates that Daugavet- and Δ-points are not equivalent in general, nor are they equivalent to their w*-versions.

ABSTRACT

A norm one element $x$ of a Banach space is a Daugavet-point (respectively,~a $Δ$-point) if every slice of the unit ball (respectively,~every slice of the unit ball containing $x$) contains an element that is almost at distance 2 from $x$. We prove the equivalence of Daugavet- and $Δ$-points in spaces of Lipschitz functions over proper metric spaces and provide two characterizations for them. Furthermore, we show that in some spaces of Lipschitz functions, there exist $Δ$-points that are not Daugavet-points. Lastly, we prove that every space of Lipschitz functions over an infinite metric space contains a $Δ$-point but might not contain any Daugavet-points.

Motivation & Objective

• To investigate the relationship between Daugavet-points and Δ-points in spaces of Lipschitz functions over metric spaces.
• To determine whether Δ-points exist in infinite-dimensional Lipschitz function spaces and under what conditions Daugavet-points arise.
• To examine the distinction between Daugavet-points, Δ-points, and their w*-versions in these function spaces.
• To provide characterizations of Δ-points in Lipschitz-free spaces over proper metric spaces, resolving Problem 2 from [13] in this context.
• To construct examples demonstrating that Daugavet-points and Δ-points are not equivalent, even in infinite-dimensional settings.

Proposed method

• Utilizes slice-based characterizations of Daugavet- and Δ-points, defining them via slices of the unit ball in the dual space.
• Applies the concept of molecules $ m_{xy} = \frac{\delta_x - \delta_y}{d(x,y)} $ in the Lipschitz-free space $ \mathcal{F}(M) $, which are dense in the unit ball.
• Employs the duality $ \mathcal{F}(M)^* = \operatorname{Lip}_0(M) $ to analyze w*-versions of Daugavet- and Δ-points.
• Constructs specific metric spaces with structured distances (e.g., involving sequences $ x_i, y_i, u_i, v_i $) to generate counterexamples.
• Uses convex combinations and support arguments to show that certain functions cannot lie in the closed convex hull of $ \Delta_\varepsilon(f) $, proving non-Δ-point status.
• Applies the inequality $ \|m_{u_{n+1}v_{n+1}} - m_{p_i q_i}\| = 1 $ to bound functional values and derive contradictions for Δ-point membership.

Experimental results

Research questions

• RQ1 Are Daugavet-points and Δ-points equivalent in spaces of Lipschitz functions over proper metric spaces?
• RQ2 Do all infinite-dimensional spaces of Lipschitz functions contain at least one Δ-point?
• RQ3 Can a space of Lipschitz functions contain Δ-points without containing any Daugavet-points?
• RQ4 Are Daugavet-points, Δ-points, and their w*-versions equivalent in $ \operatorname{Lip}_0(M) $?
• RQ5 What structural conditions on the metric space $ M $ ensure the existence of Daugavet-points in $ \operatorname{Lip}_0(M) $?

Key findings

• In spaces of Lipschitz functions over proper metric spaces, Daugavet-points and Δ-points are equivalent, and both are equivalent to their w*-versions.
• Every space of Lipschitz functions over an infinite metric space contains at least one Δ-point.
• There exist infinite-dimensional spaces of Lipschitz functions that contain Δ-points but no Daugavet-points, demonstrating that the two concepts are not equivalent.
• The paper constructs a specific example of a space of Lipschitz functions where a functional is a w*-Daugavet-point but not a Δ-point, proving that w*-Daugavet-points do not imply Δ-points.
• For any function $ f \in S_{\operatorname{Lip}_0(M)} $ that is not local, every slice of $ B_{\mathcal{F}(M)} $ defined by $ f $ contains a denting point, generalizing a result from [4].
• In the constructed example, $ f $ is a w*-Daugavet-point because for any $ \alpha > 0 $ and $ \mu \in S_{\mathcal{F}(M)} $ with finite support, a perturbation $ h $ can be found such that $ \|f - h\| = 2 $ and $ h \in S(\mu, \alpha) $, but $ f $ is not a Δ-point due to boundedness of $ g_i(m_{u_{n+1}v_{n+1}}) $ across all $ g_i \in \Delta_\varepsilon(f) $.