An Overview of L<sup>1</sup> optimal transportation on metric measure spaces

Cavalletti, Fabio

doi:10.1515/9783110550832-003

Cited by 10 publications

(2 citation statements)

References 47 publications

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“…The theory of CD/RCD spaces is now a mature topic, so wide that it would be impossible to cover it in any reasonable level of detail with a limited number of pages. Over time, various surveys have been written, see [AGS12], [AGS17], [Amb18], [Gig14], [Gig18a], [Vil], [Cav17], [Poz23] covering different aspects of the theory.…”

Section: The Focus Of This Surveymentioning

confidence: 99%

“…The technique used has little to do with those I presented in this manuscript, and is rather related to the so-called 'needle decomposition' or 'localization technique', that in some sense allows to reduce the study of some relevant geometric quantity from the original metric measure space to a suitable family of 1-dimensional metric measure spaces, where things are more tractable. For an overview on this and detailed bibliography I refer to the survey [Cav17], here I just mention that even these tools have been useful in deriving new informations about the smooth Riemannian world, see for instance [CMM19].…”

Section: Idea Of the Proofmentioning

confidence: 99%

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Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

Gigli¹,

Mondino²,

Savaré³

2015

Proc. London Math. Soc.

169

349

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The aim of this paper is to discuss convergence of pointed metric measure spaces in the absence of any compactness condition. We propose various definitions, and show that all of them are equivalent and that for doubling spaces these are also equivalent to the well-known measured Gromov-Hausdorff convergence.Then we show that the curvature conditions CD(K, ∞) and RCD(K, ∞) (Riemannian curvature dimension, RCD) are stable under this notion of convergence and that the heat flow passes to the limit as well, both in the Wasserstein and in the L 2 -framework. We also prove the variational convergence of Cheeger energies in the naturally adapted Γ-Mosco sense and the convergence of the spectra of the Laplacian in the case of spaces either uniformly bounded or satisfying the RCD(K, ∞) condition with K > 0. When applied to Riemannian manifolds, our results allow for sequences with diverging dimensions.

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Section: The Focus Of This Surveymentioning

confidence: 99%

Section: Idea Of the Proofmentioning

confidence: 99%

Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

Gigli¹,

Mondino²,

Savaré³

2015

Proc. London Math. Soc.

169

349

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The Quasi Curvature‐Dimension Condition with Applications to Sub‐Riemannian Manifolds

Milman

2020

Comm Pure Appl Math

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We obtain the best known quantitative estimates for the Lp‐Poincaré and log‐Sobolev inequalities on domains in various sub‐Riemannian manifolds, including ideal Carnot groups and in particular ideal generalized H‐type Carnot groups and the Heisenberg groups, corank 1 Carnot groups, the Grushin plane, and various H‐type foliations, Sasakian and 3‐Sasakian manifolds. Moreover, this constitutes the first time that a quantitative estimate independent of the dimension is established on these spaces. For instance, the Li‐Yau / Zhong‐Yang spectral‐gap estimate holds on all Heisenberg groups of arbitrary dimension up to a factor of 4. We achieve this by introducing a quasi‐convex relaxation of the Lott‐Sturm‐Villani CD(K, N) condition we call the “quasi curvature‐dimension condition” QCD(Q, K, N). Our motivation stems from a recent interpolation inequality along Wasserstein geodesics in the ideal sub‐Riemannian setting due to Barilari and Rizzi. We show that on an ideal sub‐Riemannian manifold of dimension n, the measure contraction property MCP(K, N) implies QCD(Q, K, N) with Q = 2N − n ≥ 1, thereby verifying the latter property on the aforementioned ideal spaces; a result of Balogh‐Kristály‐Sipos is used instead to handle nonideal corank 1 Carnot groups. By extending the localization paradigm to completely general interpolation inequalities, we reduce the study of various analytic and geometric inequalities on QCD spaces to the one‐dimensional case. Consequently, we deduce that while (strictly) sub‐Riemannian manifolds do not satisfy any type of CD condition, many of them satisfy numerous functional inequalities with exactly the same quantitative dependence (up to a factor of Q) as their CD counterparts. © 2020 Wiley Periodicals LLC

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On Objective-Based Clustering from the Perspective of Transportation Problem

Endo,

Mori

2024

Lecture Notes in Computer Science

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An Overview of L¹ optimal transportation on metric measure spaces

Cited by 10 publications

References 47 publications

Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

The Quasi Curvature‐Dimension Condition with Applications to Sub‐Riemannian Manifolds

On Objective-Based Clustering from the Perspective of Transportation Problem

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An Overview of L1 optimal transportation on metric measure spaces

Cited by 10 publications

References 47 publications

Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

The Quasi Curvature‐Dimension Condition with Applications to Sub‐Riemannian Manifolds

On Objective-Based Clustering from the Perspective of Transportation Problem

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An Overview of L¹ optimal transportation on metric measure spaces