Mass Transportation on Sub-Riemannian Manifolds

Figalli, Alessio; Rifford, Ludovic

doi:10.1007/s00039-010-0053-z

Cited by 60 publications

(97 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Unfortunately, this cannot be satisfied in more exotic topologies such as the k-holed torus (k ≥ 1), where uniqueness remains a tantalizing open question. Our discussion is predicated on global differentiability of the cost, since a wide variety of existence and uniqueness results concerning optimal solutions to the Monge-Kantorovich problem have been established for costs with singular sets-including distances in Riemannian [28,46,77], sub-Riemannian [2,5,50] and Alexandrov [11] spaces, and the mechanical actions arising from Tonelli Lagrangians [9,43]. The proof that the subtwist condition is sufficient for uniqueness relies on progress in Birkhoff's problem of characterizing extremal doubly stochastic measures on the square.…”

Section: Introductionmentioning

confidence: 99%

Optimal transportation, topology and uniqueness

2011

View full text Add to dashboard Cite

The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes Communicated by N. Trudinger. Formerly titled Extremal doubly stochastic measures and optimal transportation.It is a pleasure to thank Nassif Ghoussoub and Herbert Kellerer, who provided early encouragement in this direction, and Pierre-Andre Chiappori, Ivar Ekeland, and Lars Nesheim, whose interest in economic applications fortified our resolve to persist. We thank Wilfrid Gangbo, Jonathan Korman, and Robert Pego for fruitful discussions, Nathan Killoran for useful references, and programs of the Banff International Research Station (2003) containing the goods to be transported possess (non-trivial) topology. This question turns out to be closely linked to a delicate problem (# 111) of Birkhoff (Lattice Theory. Revised Edition, 1948): give a necessary and sufficient condition on the support of a joint probability to guarantee extremality among all measures which share its marginals. Fifty years of progress on Birkhoff's question culminate in Hestir and Williams' necessary condition which is nearly sufficient for extremality; we relax their subtle measurability hypotheses separating necessity from sufficiency slightly, yet demonstrate by example that to be sufficient certainly requires some measurability. Their condition amounts to the vanishing of the measure γ outside a countable alternating sequence of graphs and antigraphs in which no two graphs (or two antigraphs) have domains that overlap, and where the domain of each graph/antigraph in the sequence contains the range of the succeeding antigraph (respectively, graph). Such sequences are called numbered limb systems. We then explain how this characterization can be used to resolve the uniqueness of Kantorovich solutions for optimal transportation on a manifold with the topology of the sphere.

show abstract

Section: Introductionmentioning

confidence: 99%

Optimal transportation, topology and uniqueness

2011

View full text Add to dashboard Cite

show abstract

“…Estimate (1.8) gives a positive answer in our model to a question raised by Figalli and Rifford (see the Open problem at p 145-146 in [FR10]). …”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 56%

“…It is well known that a correct understanding of the cut locus is crucial in problems concerning subRiemannian optimal transport (see [AR04,AL09,FR10] and analysis of the subelliptic heat kernel (see [BBN12]). …”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

On the subRiemannian cut locus in a model of free two-step Carnot group

Montanari

Morbidelli

2017

Calc. Var.

View full text Add to dashboard Cite

We characterize the subRiemannian cut locus of the origin in the free Carnot group of step two with three generators. We also calculate explicitly the cut time of any extremal path and the distance from the origin of all points of the cut locus. Finally, by using the Hamiltonian approach, we show that the cut time of strictly normal extremal paths is a smooth explicit function of the initial velocity covector. Finally, using our previous results, we show that at any cut point the distance has a corner-like singularity.Comment: Added Section 6. Final version, to appear on Calc. Va

show abstract

“…Denote by R(·) : [0, 1] → M 2n (IR) the fundamental solution to the Cauchy probleṁ 6) and write it as…”

Section: Proof Of Theorem 22mentioning

confidence: 99%

Mass Transportation with LQ Cost Functions

2010

View full text Add to dashboard Cite

We study the optimal transport problem in the Euclidean space where the cost function is given by the value function associated with a Linear Quadratic minimization problem. Under appropriate assumptions, we generalize Brenier's Theorem proving existence and uniqueness of an optimal transport map. In the controllable case, we show that the optimal transport map has to be the gradient of a convex function up to a linear change of coordinates. We give regularity results and also investigate the non-controllable case.

show abstract

Mass Transportation on Sub-Riemannian Manifolds

Cited by 60 publications

References 33 publications

Optimal transportation, topology and uniqueness

Optimal transportation, topology and uniqueness

On the subRiemannian cut locus in a model of free two-step Carnot group

Mass Transportation with LQ Cost Functions

Contact Info

Product

Resources

About