The Monge Problem in Geodesic Spaces
We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish non branching geodesic space. We show that we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce an assumption on the transport problem π which implies that the conditional probabilities of the first marginal on each geodesic are continuous. It is known that this regularity is sufficient for the construction of an optimal transport map.
KeywordsPolish Space Transport Problem Optimal Transportation Distance Cost Geodesic Space
Unable to display preview. Download preview PDF.
- 2.S. Bianchini and L. Caravenna, On the extremality, uniqueness and optimality of transference plans. preprint.Google Scholar
- 3.S. Bianchini and F. Cavalletti, The Monge problem for distance cost in geodesic spaces, preprint (2009).Google Scholar
- 4.D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, Graduate studies in mathematics, AMS, 2001.Google Scholar
- 6.L. Caravenna, An existence result for the Monge problem in ℝn with norm cost function, preprint (2009).Google Scholar
- 7.T. Champion and L.D. Pascale, The Monge problem in ℝd, preprint (2009).Google Scholar
- 8.L. Evans and W. Gangbo, Differential equations methods for the Monge- Kantorovich mass transfer problem, Current Developments in Mathematics (1997), pp. 65–126.Google Scholar
- 10.D.H. Fremlin, Measure Theory, Vol. 4, Torres Fremlin, 2002.Google Scholar
- 12.A.M. Srivastava, A course on Borel sets, Springer, 1998.Google Scholar
- 15.C. Villani, Topics in optimal transportation, Graduate studies in mathematics, AMS, 2003.Google Scholar
- 16.––––, Optimal transport, old and new, Springer, 2008.Google Scholar