## The disintegration of the Lebesgue measure on the faces of a convex function (2009)

Citations: | 3 - 1 self |

### BibTeX

@MISC{Caravenna09thedisintegration,

author = {L. Caravenna and S. Daneri},

title = {The disintegration of the Lebesgue measure on the faces of a convex function },

year = {2009}

}

### OpenURL

### Abstract

We consider the disintegration of the Lebesgue measure on the graph of a convex function f: R n → R w.r.t. the partition into its faces, which are convex sets and therefore have a well defined linear dimension, and we prove that each conditional measure is equivalent to the k-dimensional Hausdorff measure of the k-dimensional face on which it is concentrated. The remarkable fact is that a priori the directions of the faces are just Borel and no Lipschitz regularity is known. Notwithstanding that, we

### Citations

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Citation Context ... a smooth setting. Before proving it, we devote Subsection 5.2.1 to recalls on this argument, in order to fix the notations. They are taken mainly from Chapter 4 of [Mor00] and Sections 1.5.1, 4.1 of =-=[Fed69]-=-. 5.2.1. Recalls. Let {e1, . . . , en} be a basis of R n . The wedge product between vectors is multilinear and alternating, i.e.: ( n ∑ i=1 λiei ) ∧ u1 ∧ · · · ∧ um = n∑ λi(ei ∧ u1 ∧ · · · ∧ um) i=1 ... |

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Citation Context ...me representation one would have in a smooth setting. Before proving it, we devote Subsection 5.2.1 to recalls on this argument, in order to fix the notations. They are taken mainly from Chapter 4 of =-=[Mor00]-=- and Sections 1.5.1, 4.1 of [Fed69]. 5.2.1. Recalls. Let {e1, . . . , en} be a basis of R n . The wedge product between vectors is multilinear and alternating, i.e.: ( n ∑ i=1 λiei ) ∧ u1 ∧ · · · ∧ um... |

40 | A geometrical approach to monotone functions in R n
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Citation Context .... By the convexity of f, we can moreover assume w.l.o.g. that the intersection of ∇f −1 (y) with dom ∇f is convex Since ∇f is a Borel map and Σ 1 (f) is a L n -negligible Borel set (see e.g. [AAC92], =-=[AA99]-=-), we can assume that the quotient map p of Definition 2.1 is given by ∇f and that the quotient space is given by (Im ∇f, B(Im ∇f)), which is measurably included in (R n , B(R n )). Then, this partiti... |

26 |
Geometric problems in the theory of infinitedimensional probability distributions
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Citation Context ...xpression for the conditional measures. In particular, the problem of the absolute continuity of the conditional measures w.r.t. a partition given by affine sets arose naturally in a work by Sudakov (=-=[Sud79]-=-). In absence of any Lipschitz regularity for the directions of the faces, the proof of our theorem does not rely on Area or Coarea Formula, which in several situations allow to obtain in one step bot... |

24 |
Rockafellar, Convex analysis
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Citation Context ...vex set are also called extreme points and the set of all extreme points in a convex set C will be denoted by ext(C). The definition of R-face corresponds to the definition of face of a convex set in =-=[Roc70]-=-. We also recall the following propositions, for which we refer to Section 18 of [Roc70]. Proposition 4.12. Let C = conv(D), where D is a set of points in R d , and let C ′ be a nonempty R-face of C. ... |

21 |
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Citation Context ...surable. Therefore, our result, other than answering a quite natural question, enriches the regularity properties of the faces of a convex function, which have been intensively studied for example in =-=[ELR70]-=-, [KM71], [Lar71a], [LR71], [AKP04], [PZ07]. As a byproduct, we recover the Lebesgue negligibility of the set of relative boundary points of the faces, which was first obtained in [Lar71b]. Our result... |

20 |
Monge’s transport problem on a Riemannian manifold
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Citation Context ...hich in several situations allow to obtain in one step both the existence and the absolute continuity of the disintegration (in applications to optimal mass transport problem, see for example [TW01], =-=[FM02]-=-, [AP03]). The basis of the technique we use was first presented in order to solve a variational problem in [BG07] and it has been successfully applied to the existence of optimal transport maps for s... |

19 |
Existence of optimal transport maps for crystalline norms
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Citation Context ...er than answering a quite natural question, enriches the regularity properties of the faces of a convex function, which have been intensively studied for example in [ELR70], [KM71], [Lar71a], [LR71], =-=[AKP04]-=-, [PZ07]. As a byproduct, we recover the Lebesgue negligibility of the set of relative boundary points of the faces, which was first obtained in [Lar71b]. Our result is also interesting for possible a... |

19 |
On the Monge mass transfer problem
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Citation Context ...rmula, which in several situations allow to obtain in one step both the existence and the absolute continuity of the disintegration (in applications to optimal mass transport problem, see for example =-=[TW01]-=-, [FM02], [AP03]). The basis of the technique we use was first presented in order to solve a variational problem in [BG07] and it has been successfully applied to the existence of optimal transport ma... |

11 |
On the singularities of convex functions
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Citation Context ... \ dom ∇f. By the convexity of f, we can moreover assume w.l.o.g. that the intersection of ∇f −1 (y) with dom ∇f is convex Since ∇f is a Borel map and Σ 1 (f) is a L n -negligible Borel set (see e.g. =-=[AAC92]-=-, [AA99]), we can assume that the quotient map p of Definition 2.1 is given by ∇f and that the quotient space is given by (Im ∇f, B(Im ∇f)), which is measurably included in (R n , B(R n )). Then, this... |

7 |
Disintegration and compact measures
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(Show Context)
Citation Context ...al µα(Xα) ̸= 1 (i.e. the disintegration is consistent but not strongly consistent). The more general result of existence of a disintegration which is consistent with a given partition is contained in =-=[Pac79]-=-, while a weak sufficient condition in order that a consistent disintegration is also strongly consistent is given in [HJ71]. In the following we recall an abstract disintegration theorem, in the form... |

6 |
On the Euler-Lagrange equation for a variational problem: the general case II, preprint available online at the page https://digitallibrary.sissa.it/handle/1963/2551
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Citation Context ...tegration (in applications to optimal mass transport problem, see for example [TW01], [FM02], [AP03]). The basis of the technique we use was first presented in order to solve a variational problem in =-=[BG07]-=- and it has been successfully applied to the existence of optimal transport maps for strictly convex norms in [Car08]. Just to give an idea of how this technique works, focus on a collection of 1-dime... |

5 |
Existence and stability results in the L1-theory of optimal transportation
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(Show Context)
Citation Context ...several situations allow to obtain in one step both the existence and the absolute continuity of the disintegration (in applications to optimal mass transport problem, see for example [TW01], [FM02], =-=[AP03]-=-). The basis of the technique we use was first presented in order to solve a variational problem in [BG07] and it has been successfully applied to the existence of optimal transport maps for strictly ... |

5 |
On a conjecture of Klee and Martin for convex bodies
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(Show Context)
Citation Context ...r example in [ELR70], [KM71], [Lar71a], [LR71], [AKP04], [PZ07]. As a byproduct, we recover the Lebesgue negligibility of the set of relative boundary points of the faces, which was first obtained in =-=[Lar71b]-=-. Our result is also interesting for possible applications. Indeed, the disintegration theorem is an effective tool in dimensional reduction arguments, where it may be essential to have an explicit ex... |

4 | A proof of Sudakov theorem with strictly convex norms, preprint, 2008, available at http://cvgmt.sns.it
- Caravenna
(Show Context)
Citation Context ... H n−1 ((σ t vt ) j −1 (S)), ∀ S ⊂ σ t (Z), is the flow map associated to the vector field v t j. . Indeed, if we have such a sequence of vector fields, the proof of the estimate (4.32) follows as in =-=[Car08]-=-.THE DISINTEGRATION OF THE LEBESGUE MEASURE ON THE FACES OF A CONVEX FUNCTION 17 Proof. Step 1 Preliminary considerations First of all, let us fix t ∈ [0, h +]. Eventually partitioning C into a count... |

4 |
Increasing paths on the one-skeleton of a convex body and the directions of line segments on the boundary of a convex body
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Citation Context ...ult, other than answering a quite natural question, enriches the regularity properties of the faces of a convex function, which have been intensively studied for example in [ELR70], [KM71], [Lar71a], =-=[LR71]-=-, [AKP04], [PZ07]. As a byproduct, we recover the Lebesgue negligibility of the set of relative boundary points of the faces, which was first obtained in [Lar71b]. Our result is also interesting for p... |

3 |
Sufficient conditions for optimality of c-cyclically monotone transference plans
- Bianchini, Caravenna
(Show Context)
Citation Context ... sufficient condition in order that a consistent disintegration is also strongly consistent is given in [HJ71]. In the following we recall an abstract disintegration theorem, in the form presented in =-=[BC]-=-. It guarantees, under suitable assumptions on the ambient and on the quotient measure spaces, the existence, uniqueness and strong consistency of a disintegration. Before stating it, we recall that a... |

2 |
Existence of conditional probabilities
- Hoffmann-Jorgensen
(Show Context)
Citation Context ... disintegration which is consistent with a given partition is contained in [Pac79], while a weak sufficient condition in order that a consistent disintegration is also strongly consistent is given in =-=[HJ71]-=-. In the following we recall an abstract disintegration theorem, in the form presented in [BC]. It guarantees, under suitable assumptions on the ambient and on the quotient measure spaces, the existen... |

2 |
Semicontinuity of the face-function of a convex set
- Klee
- 1971
(Show Context)
Citation Context ...Therefore, our result, other than answering a quite natural question, enriches the regularity properties of the faces of a convex function, which have been intensively studied for example in [ELR70], =-=[KM71]-=-, [Lar71a], [LR71], [AKP04], [PZ07]. As a byproduct, we recover the Lebesgue negligibility of the set of relative boundary points of the faces, which was first obtained in [Lar71b]. Our result is also... |

2 | On the points of multiplicity of monotone operators - Zajíček - 1978 |

1 |
A compact set of disjoint line segments in R3 whose end set has positive measure., Mathematika 18
- Larman
- 1971
(Show Context)
Citation Context ...ularity of the directions of these segments is not known, it may happen that the conditional measures induced by the disintegration of the Lebesgue measure are Dirac deltas (see the couterexamples in =-=[Lar71a]-=-, [AKP]). Also in our case, up to our knowledge, the directions of the faces of a convex function are just Borel measurable. Therefore, our result, other than answering a quite natural question, enric... |

1 |
On the directions of segments and r-dimensional balls on a convex surface
- Pavlica, Zajíček
(Show Context)
Citation Context ...nswering a quite natural question, enriches the regularity properties of the faces of a convex function, which have been intensively studied for example in [ELR70], [KM71], [Lar71a], [LR71], [AKP04], =-=[PZ07]-=-. As a byproduct, we recover the Lebesgue negligibility of the set of relative boundary points of the faces, which was first obtained in [Lar71b]. Our result is also interesting for possible applicati... |