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.

VL - 277 SN - 0022-1236 UR - https://www.sciencedirect.com/science/article/pii/S0022123619302289 IS - 9 JO - Journal of Functional Analysis ER - TY - JOUR T1 - The Monge Problem for Distance Cost in Geodesic Spaces JF - Communications in Mathematical Physics Y1 - 2013 A1 - Stefano Bianchini A1 - Fabio Cavalletti AB -We 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.

VL - 318 UR - https://doi.org/10.1007/s00220-013-1663-8 ER - TY - JOUR T1 - The Monge problem in Wiener space JF - Calculus of Variations and Partial Differential Equations Y1 - 2012 A1 - Fabio Cavalletti AB -We 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.

VL - 45 UR - https://doi.org/10.1007/s00526-011-0452-5 ER - TY - JOUR T1 - Optimal Transport with Branching Distance Costs and the Obstacle Problem JF - SIAM Journal on Mathematical Analysis Y1 - 2012 A1 - Fabio Cavalletti VL - 44 UR - https://doi.org/10.1137/100801433 ER - TY - CONF T1 - The Monge Problem in Geodesic Spaces T2 - Nonlinear Conservation Laws and Applications Y1 - 2011 A1 - Stefano Bianchini A1 - Fabio Cavalletti ED - Alberto Bressan ED - Chen, Gui-Qiang G. ED - Marta Lewicka ED - Wang, Dehua AB -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.

JF - Nonlinear Conservation Laws and Applications PB - Springer US CY - Boston, MA SN - 978-1-4419-9554-4 ER -