A characterisation of the Daugavet property in spaces of Lipschitz functions

García-Lirola, Luis C.; Procházka, Antonín; Zoca, Abraham Rueda

doi:10.1016/j.jmaa.2018.04.017

Cited by 42 publications

(69 citation statements)

References 21 publications

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“…This property has been recently characterized in [5] and, for a boundedly compact pointed metric space M it is shown in [ A strengthening of the concept of concavity is provided when we require all the molecules to be strongly exposed points of the unit ball of F(M ). By the characterization given in [17,Theorem 5.4], the property can be written in terms of the metric space and we may also introduce a uniform version of it. We need some notation.…”

Section: Universal Lip-bpb Property Metric Spacesmentioning

confidence: 99%

The Bishop–Phelps–Bollobás property for Lipschitz maps

Chiclana

Martı́n

2019

Nonlinear Analysis

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In this paper, we introduce and study a Lipschitz version of the Bishop-Phelps-Bollobás property (Lip-BPB property). This property deals with the possibility of making a uniformly simultaneous approximation of a Lipschitz map F and a pair of points at which F almost attains its norm by a Lipschitz map G and a pair of points such that G strongly attains its norm at the new pair of points. We first show that if M is a finite pointed metric space and Y is a finite-dimensional Banach space, then the pair (M, Y ) has the Lip-BPB property, and that both finiteness assumptions are needed. Next, we show that if M is a uniformly Gromov concave pointed metric space (i.e. the molecules of M form a set of uniformly strongly exposed points), then (M, Y ) has the Lip-BPB property for every Banach space Y . We further prove that this is the case for finite concave metric spaces, ultrametric spaces, and Hölder metric spaces. The extension of the Lip-BPB property from (M, R) to some Banach spaces Y and some results for compact Lipschitz maps are also discussed.Recently, the problem of deciding for which metric spaces M the set LipSNA(M, Y ) is (norm) dense in Lip 0 (M, Y ) has been studied. We refer the reader to [11], [16], [18], and [21], as references for this study. Let us collect here some of the known results. First, the density does not always hold, as in [21, Example 2.1] it is shown that LipSNA([0, 1], R) is not dense in Lip 0 ([0, 1], R). In fact, this can be generalized to length spaces [11, Theorem 2.2]. On the other hand, it is obvious that LipSNA(M, Y ) = Lip 0 (M, Y ) when M is finite (actually, this fact characterizes finiteness of M ). Besides, LipSNA(M, Y ) is dense in Lip 0 (M, Y ) for every Banach space Y when M is uniformly discrete, or M is countable and compact, or M is a compact Hölder metric space (i.e. M = (N, d θ ) for some metric space (N, d) and 0 < θ < 1), but

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Section: Universal Lip-bpb Property Metric Spacesmentioning

confidence: 99%

The Bishop–Phelps–Bollobás property for Lipschitz maps

Chiclana

Martı́n

2019

Nonlinear Analysis

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“…It is easy to see that one such necessary condition is that the metric segment Progress in this direction was mostly stalled until very recently. In [10], García-Lirola, Procházka and Rueda Zoca gave a complete geometric characterization of the strongly exposed points of B F (M ) (see Theorem 2.2(b)). In [1], the first author and Guirao gave a similar geometric characterization of preserved extreme points (see Theorem 2.2(a)), and asked whether extreme points could be described analogously.…”

Section: Introductionmentioning

confidence: 99%

Supports and extreme points in Lipschitz-free spaces

Aliaga

Pernecká

2020

Rev. Mat. Iberoam.

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For a complete metric space M , we prove that the finitely supported extreme points of the unit ball of the Lipschitz-free space F(M ) are precisely the elementary molecules (δ(p) − δ(q))/d(p, q) defined by pairs of points p, q in M such that the triangle inequality d(p, q) < d(p, r) + d(q, r) is strict for any r ∈ M different from p and q. To this end, we show that the class of Lipschitz-free spaces over closed subsets of M is closed under arbitrary intersections when M has finite diameter, and that this allows a natural definition of the support of elements of F(M ).

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“…f x,y (t) := d(x, y) 2 d(t, y) − d(t, x) d(t, y) + d(t, x)for every t ∈ M , and take h x,y = 1 2 (g x,y + f x,y ). Now, one can check that the family B = {h x,y } (x,y)∈Λ does the work following word-by-word the proof of[21, Theorem 5.4].…”

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confidence: 99%

On strongly norm attaining Lipschitz maps

Cascales

Chiclana

García-Lirola

et al. 2019

Journal of Functional Analysis

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We study the set SNA(M, Y ) of those Lipschitz maps from a (complete pointed) metric space M to a Banach space Y which (strongly) attain their Lipschitz norm (i.e. the supremum defining the Lipschitz norm is a maximum). Extending previous results, we prove that this set is not norm dense when M is a length space (or local) or when M is a closed subset of R with positive Lebesgue measure, providing new examples which have very different topological properties than the previously known ones. On the other hand, we study the linear properties which are sufficient to get Lindenstrauss property A for the Lipschitz-free space F (M ) over M , and show that all of them actually provide the norm density of SNA(M, Y ) in the space of all Lipschitz maps from M to any Banach space Y . Next, we prove that SNA(M, R) is weakly sequentially dense in the space of all Lipschitz functions for all metric spaces M . Finally, we show that the norm of the bidual space of F (M ) is octahedral provided the metric space M is discrete but not uniformly discrete or M is infinite.J. Lindenstrauss extended such study to general linear operators, showed that this is not always possible, and also gave positive results. If we say that a Banach space X has (Lindenstrauss) property A when NA(X, Y ) = L(X, Y ) for every Banach space Y , it is shown in [36] that reflexive spaces have property A. This result was extended by J. Bourgain [9] showing that Banach spaces X with the RNP also have Lindenstrauss property A, and he also provided a somehow reciprocal result. We refer the interested reader to the survey paper [3] for a detailed account on norm attaining linear operators.

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A characterisation of the Daugavet property in spaces of Lipschitz functions

Cited by 42 publications

References 21 publications

The Bishop–Phelps–Bollobás property for Lipschitz maps

The Bishop–Phelps–Bollobás property for Lipschitz maps

Supports and extreme points in Lipschitz-free spaces

On strongly norm attaining Lipschitz maps

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