## THE MONGE PROBLEM FOR DISTANCE COST IN GEODESIC SPACES

Citations: | 6 - 2 self |

### BibTeX

@MISC{Bianchini_themonge,

author = {Stefano Bianchini and Fabio Cavalletti},

title = {THE MONGE PROBLEM FOR DISTANCE COST IN GEODESIC SPACES},

year = {}

}

### OpenURL

### Abstract

Abstract. We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dL is a geodesic Borel distance which makes (X, dL) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem π which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1-dimensional Hausdorff distance induced by dL. It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show

### Citations

88 | Gangbo: Differential equations methods for the Monge-Kantorovich mass transfer problem
- Evans, W
- 1999
(Show Context)
Citation Context ... the property claimed by Sudakov is not true. An exemple with d = 3, k = 1 can be found in [13]. The Euclidean case has been correctly solved only during the last decade. L. C. Evans and W. Gangbo in =-=[9]-=- solved the problem under the assumptions that sptµ ∩ sptν = ∅, µ, ν ≪ L d and their densities are Lipschitz function with compact support. The first existence results for general absolutely continuou... |

84 |
On the geometry of metric measure spaces
- Sturm
(Show Context)
Citation Context ... 8.2. Approximations by metric spaces. In this section we explain a procedure to verify if the transport problem under consideration satisfied Assumption 2. The basic references for this sections are =-=[16, 17]-=-. We consider the following setting: (1) (Xn, dn, dL,n), n ∈ N, is a family of metric structures satisfying the assumptions of page 8 and Remark 2.13; (2) µn, νn ∈ P(Xn), µn ⊥ νn; (3) πn ∈ Π(µn, νn) d... |

69 |
Kirchheim: Currents in metric spaces
- Ambrosio, B
- 2000
(Show Context)
Citation Context ...e (S, S , η) induced by a map r. From the above definition the map r is clearly measurable and inverse measure preserving. Definition 2.2. A disintegration of ρ consistent with r is a map ρ : R × S → =-=[0, 1]-=- such that (1) ρs(·) is a probability measure on (R, R) for all s ∈ S, (2) ρ·(B) is η-measurable for all B ∈ R,6 STEFANO BIANCHINI AND FABIO CAVALLETTI and satisfies for all B ∈ R, C ∈ S the consiste... |

36 |
Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs
- Caffarelli, Feldman, et al.
(Show Context)
Citation Context ...with compact support. The first existence results for general absolutely continuous measures µ, ν with compact support have been independently obtained by L. Caffarelli, M. Feldman and R.J. McCann in =-=[6]-=- and by N. Trudinger and X.J. Wang in [19]. Afterwards M. Feldman and R.J. McCann [10] extended the results to manifolds with geodesic cost. The case of a general norm as cost function on R d , includ... |

31 |
A course on Borel sets
- Srivastava
- 1998
(Show Context)
Citation Context ... such that A = P1(B). The coprojective class Π1 n+1 (X) is the complement in X of the class Σ1 n+1. If Σ1 n , Π1n are the projective, coprojective pointclasses, then the following holds (Chapter 4 of =-=[15]-=-): (1) Σ1 n , Π1n are closed under countable unions, intersections (in particular they are monotone classes); (2) Σ1 n is closed w.r.t. projections, Π1 n is closed w.r.t. coprojections; (3) if A ∈ Σ1 ... |

22 |
A Course in Metric Geometry, Graduate studies in mathematics
- Burago, Burago, et al.
(Show Context)
Citation Context ...that there exists an R-measurable section f: this measurability condition implies that f is constant on atoms, in particular on equivalence classes. □ 2.4. Metric setting. In this section we refer to =-=[5]-=-. Definition 2.8. A length structure on a topological space X is a class A of admissible paths, which is a subset of all continuous paths in X, together with a map L : A → [0, +∞]: the map L is called... |

20 |
Monge’s transport problem on a Riemannian manifold
- Feldman, McCann
- 2002
(Show Context)
Citation Context ...easures µ, ν with compact support have been independently obtained by L. Caffarelli, M. Feldman and R.J. McCann in [6] and by N. Trudinger and X.J. Wang in [19]. Afterwards M. Feldman and R.J. McCann =-=[10]-=- extended the results to manifolds with geodesic cost. The case of a general norm as cost function on R d , including also the case with non strictly convex unitary ball, has been solved first in the ... |

19 | Existence of optimal transport maps for crystalline norms - Ambrosio, Kirchheim, et al. |

19 |
On the Monge mass transfer problem
- Trudinger, Wang
- 2001
(Show Context)
Citation Context ... results for general absolutely continuous measures µ, ν with compact support have been independently obtained by L. Caffarelli, M. Feldman and R.J. McCann in [6] and by N. Trudinger and X.J. Wang in =-=[19]-=-. Afterwards M. Feldman and R.J. McCann [10] extended the results to manifolds with geodesic cost. The case of a general norm as cost function on R d , including also the case with non strictly convex... |

17 |
On the measure contraction property of metric measure spaces
- Ohta
(Show Context)
Citation Context ...ssumption 2 holds. So if d is l.s.c. and (X, d, µ) satisfies MCP(K, N) we have a solution of the Monge problem. For a complete view on the so colled Measure Contraction Property MCP(K, N) we refer to =-=[14]-=- and [12]. Observe that the Heisenberg group and Alexandrov spaces satisfy MCP.4 STEFANO BIANCHINI AND FABIO CAVALLETTI 1.2. Structure of the paper. The paper is organized as follows. In Section 2, w... |

6 |
The Monge problem
- Champion, Pascale
(Show Context)
Citation Context ... the particular case of crystalline norm by L. Ambrosio, B. Kirchheim and A. Pratelli in [2], and then in fully generality independently by L. Caravenna in [7] and by T. Champion and L. De Pascale in =-=[8]-=-. 1.1. Overview of the paper. The presence of 1-dimensional sets (the geodesics) along which the cost is linear is a strong degeneracy for transport problems. This degeneracy is equivalent to the foll... |

6 |
On the geometry of metric measure spaces II
- Sturm
(Show Context)
Citation Context ... 8.2. Approximations by metric spaces. In this section we explain a procedure to verify if the transport problem under consideration satisfied Assumption 2. The basic references for this sections are =-=[16, 17]-=-. We consider the following setting: (1) (Xn, dn, dL,n), n ∈ N, is a family of metric structures satisfying the assumptions of page 8 and Remark 2.13; (2) µn, νn ∈ P(Xn), µn ⊥ νn; (3) πn ∈ Π(µn, νn) d... |

5 |
An existence result for the Monge problem in R n with norm cost functions. Preprint at http://cvgmt.sns.it/cgi/get.cgi/papers/cara/sel.partMonge.pdf
- Caravenna
(Show Context)
Citation Context ...onvex unitary ball, has been solved first in the particular case of crystalline norm by L. Ambrosio, B. Kirchheim and A. Pratelli in [2], and then in fully generality independently by L. Caravenna in =-=[7]-=- and by T. Champion and L. De Pascale in [8]. 1.1. Overview of the paper. The presence of 1-dimensional sets (the geodesics) along which the cost is linear is a strong degeneracy for transport problem... |

4 |
Measure Theory, volume 4
- Fremlin
- 2002
(Show Context)
Citation Context ...here exists a countably generated σ-algebra ˆ H such that for all A ∈ H there exists Â ∈ ˆ H such that m(A △ Â) = 0. We recall the following version of the disintegration theorem that can be found on =-=[11]-=-, Section 452 (see [3] for a direct proof). Theorem 2.3 (Disintegration of measures). Assume that (R, R, ρ) is a countably generated probability space, R = ∪s∈SRs a partition of R, r : R → S the quoti... |

4 |
A compact set of disjoint line segments in R 3 whose end set has positive measure
- Larman
- 1971
(Show Context)
Citation Context ...ous with respect to the H k measure of the correct dimension. But it turns out that when d > 2, 0 < k < d − 1 the property claimed by Sudakov is not true. An exemple with d = 3, k = 1 can be found in =-=[13]-=-. The Euclidean case has been correctly solved only during the last decade. L. C. Evans and W. Gangbo in [9] solved the problem under the assumptions that sptµ ∩ sptν = ∅, µ, ν ≪ L d and their densiti... |

3 |
Ricci curvature bounds and geometric inequalities in the Heisenberg group. preprint SFB611
- Juillet
- 2006
(Show Context)
Citation Context ... 2 holds. So if d is l.s.c. and (X, d, µ) satisfies MCP(K, N) we have a solution of the Monge problem. For a complete view on the so colled Measure Contraction Property MCP(K, N) we refer to [14] and =-=[12]-=-. Observe that the Heisenberg group and Alexandrov spaces satisfy MCP.4 STEFANO BIANCHINI AND FABIO CAVALLETTI 1.2. Structure of the paper. The paper is organized as follows. In Section 2, we recall ... |

2 | On the extremality, uniqueness and optimality of transference plans
- Bianchini, Caravenna
(Show Context)
Citation Context ... Theorem 1.2 holds. To conclude this introduction, we observe that it is probably possible to extend these results to the case −dL analytic function on (X × X, d × d), see for example the analysis of =-=[3]-=-. We note moreover that in the case d = dL and (X, d, µ) geodesic measure space satisfying the MCP(K, N) condition for some K, N real numbers, N ≥ 1, we can prove, via Theorem 8.6, that Assumption 2 h... |

2 |
Geometric problems in the theory of dimensional distributions
- Sudakov
- 1979
(Show Context)
Citation Context ... the Euclidean norm and the measures µ, ν were supposed to be absolutely continuous and supported on two disjoint compact sets. The original problem remained unsolved for a long time. In 1978 Sudakov =-=[18]-=- claimed to have a solution for any distance cost function induced by a norm: an essential ingredient in the proof was that if µ ≪ L d and L d -a.e. R d can be decomposed into convex sets of dimension... |

1 |
The Euler-Lagrange equation for a singular variational problem
- Bianchini, Gloyer
- 2009
(Show Context)
Citation Context ... converge pointwise to continuous functions. A special case is when dL = d: a natural approximation is by transport plans where ν is atomic, with a finite number of atoms. Using techniques similar to =-=[4]-=-, one can prove the next proposition. Proposition 8.8. Let {xn}n∈N be a dense sequence in X, and assume that for all transport problems {∫ } inf d(x, y)π, π ∈ Π(µ, δxn) the disintegration along transp... |