Smooth semi-Lipschitz functions and almost isometries between Finsler manifolds

Daniilidis, Aris; Jaramillo, Jesús Á.; Venegas, M Francisco

doi:10.1016/j.jfa.2020.108662

Cited by 12 publications

(16 citation statements)

References 9 publications

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“…Two quasi-metric spaces can be completely identified via isometries. (The reader should be alerted that the slightly weaker notion of almost isometry also exists, and is more appropriate in relation to Banach-Stone type theorems [9,14].) Definition 2.4 (Isometry).…”

Section: Remark 23 (Terminology Alert I)mentioning

confidence: 99%

“…Remark 2.31 (Terminology alert III). The above definition of a semi-Lipschitz function, introduced in [14], differs from the one that is usually considered in the literature and is based on an inequality of the form (2.12) f (x) − f (y) ≤ Ld(x, y).…”

Section: Semi-lipschitz Functions and Dual Conesmentioning

confidence: 99%

“…In particular, the class of Lipschitz functions is clearly affected if we consider asymmetric distances or asymmetrizations of the space. In both cases semi-Lipschitz functions are appropriate morphisms to describe properties of the space (see [14] e.g.) and this work outlines a natural way to define a notion of a quasi-metric free space as well as of a canonical asymmetrization of a space.…”

Section: Properties (S) and (S *mentioning

confidence: 99%

“…Quasi-metric spaces and asymmetric norms have recently attracted a lot of interest: they arise naturally when considering non-reversible Finsler manifolds [9,14,33] (see also [7,15]), and have applications in physics [25], as well as in game theory [1,19]. The properties of spaces with asymmetric norms have been studied by several authors (see [11,36] and references therein), emphasizing similarities and differences with respect to the theory of (symmetric) normed spaces.…”

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confidence: 99%

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Asymmetric free spaces and canonical asymmetrizations

2021

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A construction analogous to that of Godefroy-Kalton for metric spaces allows one to embed isometrically, in a canonical way, every quasi-metric space (X, d) in an asymmetric normed space Fa(X, d) (its quasi-metric free space, also called asymmetric free space or semi-Lipschitz free space). The quasi-metric free space satisfies a universal property (linearization of semi-Lipschitz functions). The (conic) dual of Fa(X, d) coincides with the non-linear asymmetric dual of (X, d), that is, the space SLip 0 (X, d) of semi-Lipschitz functions on (X, d), vanishing at a base point. In particular, for the case of a metric space (X, D), the above construction yields its usual free space. On the other hand, every metric space (X, D) naturally inherits a canonical asymmetrization coming from its free space F(X). This gives rise to a quasi-metric space (X, D+) and an asymmetric free space Fa(X, D+). The symmetrization of the latter is isomorphic to the original free space F(X). The results of this work are illustrated with explicit examples.

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Section: Remark 23 (Terminology Alert I)mentioning

confidence: 99%

Section: Semi-lipschitz Functions and Dual Conesmentioning

confidence: 99%

Section: Properties (S) and (S *mentioning

confidence: 99%

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confidence: 99%

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Asymmetric free spaces and canonical asymmetrizations

2021

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“…to [8,9], [11], [2,3] and [15]. Quasi-metric spaces and asymmetric norms have recently attracted a lot of interest in modern mathematics, they arise naturally when considering non-reversible Finsler manifolds [10,5,13]. For an introduction and study of asymmetric free spaces (or semi-Lipschitz free spaces), we refer to the recent paper [6].…”

Section: Introductionmentioning

confidence: 99%