Points of differentiability of the norm in Lipschitz-free spaces

Aliaga, Ramón J.; Zoca, Abraham Rueda

doi:10.1016/j.jmaa.2020.124171

Cited by 7 publications

(3 citation statements)

References 21 publications

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“…Note that given u, v ∈ M , then u, v satisfy that [u, v] = {u, v} if, and only if, m u,v is an extreme point of B F (M ) [5, Theorem 3.2], which is in turn equivalent to be a preserved extreme point since M is compact [4, Theorem 4.2], which in turn is equivalent to being a denting point by [12,Theorem 2.4]. Moreover, by [6,Theorem 2.4] we get that m x,y ± m u,v = 2 is equivalent to the inequality d(x, y)…”

Section: Daugavet Pointsmentioning

confidence: 99%

Daugavet points and $Δ$-points in Lipschitz-free spaces

Jung¹,

Zoca²

2020

Preprint

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We study Daugavet points and ∆-points in Lipschitz-free Banach spaces. We prove that, if M is a compact metric space, then µ ∈ S F (M ) is a Daugavet point if, and only if, there is no denting point of B F (M ) at distance strictly smaller than two from µ. Moreover, we prove that if x and y are connectable by rectifiable curves of lenght as close to d(x, y) as we wish, then the molecule m x,y is a ∆-point. Some conditions on M which guarantee that the previous implication reverses are also obtained. As a consequence of our work, we show that Lipschitz-free spaces are natural examples of Banach spaces where we can guarantee the existence of ∆-points which are not Daugavet points.

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Section: Daugavet Pointsmentioning

confidence: 99%

Daugavet points and $Δ$-points in Lipschitz-free spaces

Jung¹,

Zoca²

2020

Preprint

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“…Let us first introduce some notation for segments and approximate segments (see also [5]). For all u, v ∈ M and δ > 0 let…”

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confidence: 99%

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

Veeorg

2023

Studia Math.

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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.

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“…However, this is in general not possible. We refer to [8] for counterexamples and necessary conditions.…”

Section: Elementary Moleculesmentioning

confidence: 99%

Geometry and structure of Lipschitz-free spaces and their biduals

Varea¹,

Ramón²

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Lipschitz-free spaces F(M ) are canonical linearizations of arbitrary complete metric spaces M . More specifically, F(M ) is the unique Banach space that contains an isometric copy of M that is linearly dense, and such that any Lipschitz mapping from M into some Banach space X extends to a bounded linear operator from F(M ) into X. Those spaces are a very powerful tool for studies of the nonlinear geometry of Banach spaces, as they allow the application of well-known classical linear techniques to nonlinear problems. But this effort is only worthwhile if we have sufficient knowledge about the structure of F(M ). The systematic study of Lipschitz-free spaces is rather recent and so the current understanding of their structure is still quite limited. This thesis is framed within the general program of studying the structure of general Lipschitz-free spaces.We start our study by developing some basic tools for the general theory of Lipschitz-free spaces. First we introduce weighting operators and use them to solve Weaver's conjecture that all normal functionals in the bidual F(M ) * * are weak * continuous. Next we prove the intersection theorem, which essentially says that the intersection of Lipschitz-free spaces is again a Lipschitz-free space. That result allows us to develop the concept of support of an element of F(M ), analogous to the support of a measure. Furthermore, we extend the use of these tools to the bidual F(M ) * * and apply them to establish a decomposition of the bidual into spaces of functionals that are "concentrated at infinity" and "separated from infinity", respectively. * * que poden expressar-se com a diferència de dos elements positius, com ara l'existència d'un anàleg de la descomposició de Jordan per a mesures.En segon lloc, estudiem l'estructura extremal de la bola unitat de F(M ) i fem algunes contribucions al programa general consistent en trobar caracteritzacions purament geomètriques de tots els seus elements extremals. Concretament, caracteritzem els punts extrems preservats de la bola, així com aquells punts extrems i exposats que tenen suport finit. A més fem una descripció completa de l'estructura extremal de la part positiva de la bola unitat. La teoria dels suports en F(M ) desenvolupada anteriorment juga un paper crucial en les demostracions d'aquests resultats.viii

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Points of differentiability of the norm in Lipschitz-free spaces

Cited by 7 publications

References 21 publications

Daugavet points and $Δ$-points in Lipschitz-free spaces

Daugavet points and $Δ$-points in Lipschitz-free spaces

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

Geometry and structure of Lipschitz-free spaces and their biduals

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