We prove that if (X,d,m) is an essentially non-branching metric measure space with m(X)=1, having Ricci curvature bounded from below by K and dimension bounded above by N∈(1,∞), understood as a synthetic condition called Measure-Contraction property, then a sharp isoperimetric inequality à la Lévy-Gromov holds true. Measure theoretic rigidity is also obtained.

%V 277 %P 2893 - 2917 %8 2019/11/01/ %@ 0022-1236 %G eng %U https://www.sciencedirect.com/science/article/pii/S0022123619302289 %N 9 %! Journal of Functional Analysis %0 Journal Article %J Communications in Mathematical Physics %D 2013 %T The Monge Problem for Distance Cost in Geodesic Spaces %A Stefano Bianchini %A Fabio Cavalletti %XWe address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dLis 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 that in this setting dL-cyclical monotonicity is not sufficient for optimality.

%B Communications in Mathematical Physics %V 318 %P 615–673 %8 Mar %G eng %U https://doi.org/10.1007/s00220-013-1663-8 %R 10.1007/s00220-013-1663-8 %0 Journal Article %J Calculus of Variations and Partial Differential Equations %D 2012 %T The Monge problem in Wiener space %A Fabio Cavalletti %XWe address the Monge problem in the abstract Wiener space and we give an existence result provided both marginal measures are absolutely continuous with respect to the infinite dimensional Gaussian measure γ.

%B Calculus of Variations and Partial Differential Equations %V 45 %P 101–124 %8 Sep %G eng %U https://doi.org/10.1007/s00526-011-0452-5 %R 10.1007/s00526-011-0452-5 %0 Journal Article %J SIAM Journal on Mathematical Analysis %D 2012 %T Optimal Transport with Branching Distance Costs and the Obstacle Problem %A Fabio Cavalletti %B SIAM Journal on Mathematical Analysis %V 44 %P 454-482 %G eng %U https://doi.org/10.1137/100801433 %R 10.1137/100801433 %0 Conference Paper %B Nonlinear Conservation Laws and Applications %D 2011 %T The Monge Problem in Geodesic Spaces %A Stefano Bianchini %A Fabio Cavalletti %E Alberto Bressan %E Chen, Gui-Qiang G. %E Marta Lewicka %E Wang, Dehua %XWe 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.

%B Nonlinear Conservation Laws and Applications %I Springer US %C Boston, MA %P 217–233 %@ 978-1-4419-9554-4 %G eng