A rigidity theorem in Alexandrov spaces with lower curvature bound

Yokota, Takumi

doi:10.1007/s00208-011-0686-8

Cited by 13 publications

(14 citation statements)

References 17 publications

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“…Remark 5.6. Note that Theorem 45 of [Yok12] requires the tangent cone to be separable. It is not clear to us whether the tangent cone of a separable geodesic space with curvature bounded below is always separable.…”

Section: Proof Of Theorem 32mentioning

confidence: 99%

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Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics

Ahidar-Coutrix

Gouic

Paris

2019

Probab. Theory Relat. Fields

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This paper provides rates of convergence for empirical (generalised) barycenters on compact geodesic metric spaces under general conditions using empirical processes techniques. Our main assumption is termed a variance inequality and provides a strong connection between usual assumptions in the field of empirical processes and central concepts of metric geometry. We study the validity of variance inequalities in spaces of non-positive and non-negative Aleksandrov curvature. In this last scenario, we show that variance inequalities hold provided geodesics, emanating from a barycenter, can be extended by a constant factor. We also relate variance inequalities to strong geodesic convexity. While not restricted to this setting, our results are largely discussed in the context of the 2-Wasserstein space.Given a separable and complete metric space (M, d), define P 2 (M ) as the set of Borel probability measures P on M such that). (1.1) When it exists, a barycenter stands as a natural analog of the mean of a (square integrable) probability measure on R d . Alternative notions of mean value include local minimisers [Kar14], p-means [Yok17], exponential barycenters [ÉM91] or convex means [ÉM91]. Extending the notion of mean value to the case of probability measures on spaces M with no Euclidean (or Hilbert) structure has a number of applications ranging from geometry [Stu03] and optimal transport [Vil03, Vil08, San15, CP19] to statistics and data science [Pel05, BLL15, BGKL18, KSS19], and the context of abstract metric spaces provides a unifying framework encompassing many non-standard settings. Properties of barycenters, such as existence and uniqueness, happen to be closely related to geometric characteristics of the space M . These properties are addressed in the context of Riemannian manifolds in [Afs11]. Many interesting examples of metric spaces, however, cannot be described as smooth manifolds because of their singularities or infinite dimensional nature. More general geometrical structures are geodesic metric spaces which include many more examples of interest (precise definitions and necessary background on metric geometry are reported in Appendix A). The barycenter problem has been addressed in this general setting. The scenario where M has non-positive curvature (from here on, curvature bounds are understood in the sense of Aleksandrov) is considered in [Stu03]. More generally, the case of metric spaces with upper bounded curvature is studied in [Yok16] and [Yok17]. The context of spaces M with lower bounded curvature is discussed in [Yok12] and [Oht12].Focus on the case of metric spaces with non-negative curvature may be motivated by the increasing interest for the theory of optimal transport and its applications. Indeed, a space of central importance in this context is the Wasserstein space M = P 2 (R d ), equipped with the Wasserstein metric W 2 , known to be geodesic and with non-negative curvature (see Section 7.3 in [AGS08]). In this framework, the barycenter problem was first studied by [AC11] and has si...

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Section: Proof Of Theorem 32mentioning

confidence: 99%

“…• Whose metric d p is defined (without ambiguity) by Note that the tangent space is not always a geodesic space, see discussion before Proposition 28 in [Yok12] and the Proposition itself for the proof. Notation .…”

Section: A5 Tangent Conesmentioning

confidence: 99%

Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics

Ahidar-Coutrix

Gouic

Paris

2019

Probab. Theory Relat. Fields

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“…Put Λ := (log z ) µ and note that Lemma 4.6 yields Cz×Cz u, v z dΛ(u)dΛ(v) = 0. Then, as supp Λ is separable, Yokota's theorem [Yo,Theorems A,27] can be applied and shows that supp Λ is contained in a subset which is isometric to a Hilbert space.…”

Section: Barycenters At the Origins Of Tangent Conesmentioning

confidence: 99%

Barycenters in Alexandrov spaces of curvature bounded below

Ohta

2012

Advances in Geometry

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“…We present here a useful tool for the study of barycenters on Alexandrov spaces with curvature bounded below: the support of log b #P in the tangent cone at the barycenter is included in a Hilbert space. This rigidity result has been stated in [9] as Theorem 45, however the proof is not written. Moreover, there is an extra assumption of support of log b #P being separable, which does not even seem to be a consequence of the support of P being separable.…”

Section: Introductionmentioning

confidence: 99%

“…For measurability purposes (see Lemma 6), we suppose however that S is separable. The proof is essentially the one of Theorem 45 of [9], with needed approximations dealt with a bit differently.…”

Section: Introductionmentioning

confidence: 99%

A note on flatness of non separable tangent cone at a barycenter

Gouic¹

2020

Comptes Rendus. Mathématique

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Given a probability measure P on an Alexandrov space S with curvature bounded below, we prove that the support of the pushforward of P on the tangent cone T b S at its (exponential) barycenter b is a subset of a Hilbert space, without separability of the tangent cone. Résumé. Étant donné une mesure de probabilité P sur un espace d'Alexandrov S avec courbure minorée, nous prouvons que le support de la mesure poussée de P sur le cône tangent T b S à son barycentre (exponentiel) b est un sous-ensemble d'un espace de Hilbert, sans condition de séparabilité du cône tangent.

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A rigidity theorem in Alexandrov spaces with lower curvature bound

Cited by 13 publications

References 17 publications

Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics

Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics

Barycenters in Alexandrov spaces of curvature bounded below

A note on flatness of non separable tangent cone at a barycenter

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