The disintegration of the Lebesgue measure on the faces of a convex function

Caravenna, Laura; Daneri, Sara

doi:10.1016/j.jfa.2010.01.024

Cited by 16 publications

(20 citation statements)

References 15 publications

(29 reference statements)

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“…In this section we recall, in an abstract and more general setting, the main steps of the disintegration technique first introduced in [8] for partitions into segments and then extended to locally affine partitions of any dimension in [13]. This technique allows to prove the absolute continuity of the conditional probabilities of the Lebesgue measure and to deduce that the initial and final points of a directed locally affine partition are Lebesgue negligible, provided the direction map satisfies a suitable regularity assumption that we call (initial/final ) forward/backward cone approximation property.…”

Section: Dimensional Reduction On Directed Partitions Via Cone Approxmentioning

confidence: 99%

On Sudakov’s type decomposition of transference plans with norm costs

Daneri

2018

Memoirs of the AMS

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Abstract. We consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost | · | D * min |T(x) − x| D * dµ(x), T :with µ, ν probability measures in R d and µ absolutely continuous w.r.t. L d . The key idea in this approach is to decompose (via disintegration of measures) the Kantorovich optimal transportation problem into a family of transportation problems in Za × R d , where {Za} a∈A ⊂ R d are disjoint regions such that the construction of an optimal map Ta : Za → R d is simpler than in the original problem, and then to obtain T by piecing together the maps Ta. When the norm | · | D * is strictly convex [25], the sets Za are a family of 1-dimensional segments determined by the Kantorovich potential called optimal rays, while the existence of the map Ta is straightforward provided one can show that the disintegration of L d (and thus of µ) on such segments is absolutely continuous w.r.t. the 1-dimensional Hausdorff measure [12]. When the norm | · | D * is not strictly convex, the main problems in this kind of approach are two: first, to identify a suitable family of regions {Za} a∈A on which the transport problem decomposes into simpler ones, and then to prove the existence of optimal maps. In this paper we show how these difficulties can be overcome, and that the original idea of Sudakov can be successfully implemented.The results yield a complete characterization of the Kantorovich optimal transportation problem, whose straightforward corollary is the solution of the Monge problem in each set Za and then in R d . The strategy is sufficiently powerful to be applied to other optimal transportation problems.The analysis requires (1) the study of the transportation problem on directed locally affine partitionswhich are relatively open in their k-dimensional affine hull and on which the transport occurs only along directions belonging to a cone C k a ; (2) the proof of the absolute continuity w.r.t. the suitable k-dimensional Hausdorff measure of the disintegration of L d on these directed locally affine partitions; (3) the definition of cyclically connected sets w.r.t. a family of transportation plans with finite cone costs; (4) the proof of the existence of cyclically connected directed locally affine partitions for transport problems with cost functions which are indicator functions of cones and no potentials can be constructed.

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Section: Dimensional Reduction On Directed Partitions Via Cone Approxmentioning

confidence: 99%

On Sudakov’s type decomposition of transference plans with norm costs

Daneri

2018

Memoirs of the AMS

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“…This is a consequence of Rohlin disintegration theorem (see [14], [2] for a recent version and references therein).…”

Section: The Occupation Time Formula 21 Disintegration Of Random Meamentioning

confidence: 83%

An occupation time formula for semimartingales inRN

Bevilacqua

Flandoli

2014

Stochastic Processes and their Applications

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Inspired by coarea formula in geometric measure theory, an occupation time formula for continuous semimartingales in R N is proven. The occupation measure of a semimartingale, for N ≥ 2, is singular with respect to Lebesgue measure but it has a bounded density "transversal" to a foliation, under proper assumptions. In the particular case of the foliation given locally by the distance function from a manifold, the transversal density is related to a geometric local time of the semimartingale at the manifolds of the foliation. R Γa f (x) Q i A,φ (a, dx) ν i A,φ (da)

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“…In [14] Caravenna has carried out the original strategy of Sudakov for general strictly convex norms and eventually Bianchini and Daneri in [10] accomplished the plan of a proof of Sudakov for general norms on finitedimensional normed spaces. Let us note here also a paper [15], which deals with a related problem in the context of faces of convex functions.…”

Section: Optimal Transportmentioning

confidence: 99%

Leaves decompositions in Euclidean spaces

Ciosmak¹

2021

Preprint

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We partly extend the localisation technique from convex geometry to the multiple constraints setting. For a given 1-Lipschitz map u : R n → R m , m ≤ n, we define and prove the existence of a partition of R n , up to a set of Lebesgue measure zero, into maximal closed convex sets such that restriction of u is an isometry on these sets.We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost every set of the partition of dimension m, the associated conditional measure is log-concave. This result is proven also in the context of the curvature-dimension condition for weighted Riemannian manifolds. This partially confirms a conjecture of Klartag. RésuméNous étendons en partie la technique de localisation de la géométrie convexe pour plusieurs contraintes. Étant donnée application 1-lipschitzienne u : R n → R m , m ≤ n, nous définissons et prouvons l'existence d'une partition de R n , en dehors d'un ensemble négligeable, à ensembles convexes fermés maximaux tels que la restriction de u soit une isométrie sur ces ensembles.On considère une désintégration, relativement à cette partition, d'une mesure log-concave. Nous montrons que pour presque chaque ensemble m-dimensionnelle de la partition, la mesure conditionnelle associée est log-concave. Ce résultat est également prouvé dans le contexte de la condition de courbure-dimension pour les variétés riemanniennes pondérées. Cela confirme en partie une conjecture de Klartag.

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The disintegration of the Lebesgue measure on the faces of a convex function

Cited by 16 publications

References 15 publications

On Sudakov’s type decomposition of transference plans with norm costs

On Sudakov’s type decomposition of transference plans with norm costs

An occupation time formula for semimartingales inRN

Leaves decompositions in Euclidean spaces

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