@article {2021, title = {Displacement convexity of Entropy and the distance cost Optimal Transportation}, journal = {Annales de la Facult{\'e} des sciences de Toulouse : Math{\'e}matiques}, volume = {Ser. 6, 30}, year = {2021}, pages = {411{\textendash}427}, doi = {10.5802/afst.1679}, url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1679/}, author = {Fabio Cavalletti and Nicola Gigli and Flavia Santarcangelo} } @article {2021, title = {Independence of synthetic curvature dimension conditions on transport distance exponent}, journal = {Trans. Amer. Math. Soc.}, volume = {374}, year = {2021}, pages = {5877{\textendash}5923}, issn = {0002-9947}, doi = {10.1090/tran/8413}, url = {https://doi.org/10.1090/tran/8413}, author = {Afiny Akdemir and Andrew Colinet and Robert McCann and Fabio Cavalletti and Flavia Santarcangelo} } @booklet {2020, title = {Indeterminacy estimates and the size of nodal sets in singular spaces}, year = {2020}, keywords = {Differential Geometry (math.DG), FOS: Mathematics, Metric Geometry (math.MG)}, doi = {10.48550/ARXIV.2011.04409}, url = {https://arxiv.org/abs/2011.04409}, author = {Fabio Cavalletti and Sara Farinelli} } @article {2019, title = {Isoperimetric inequality under Measure-Contraction property}, volume = {277}, year = {2019}, month = {2019/11/01/}, pages = {2893 - 2917}, abstract = {

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,$\infty$), understood as a synthetic condition called Measure-Contraction property, then a sharp isoperimetric inequality {\`a} la L{\'e}vy-Gromov holds true. Measure theoretic rigidity is also obtained.

}, keywords = {Isoperimetric inequality, Measure-Contraction property, Optimal transport, Ricci curvature}, isbn = {0022-1236}, url = {https://www.sciencedirect.com/science/article/pii/S0022123619302289}, author = {Fabio Cavalletti and Flavia Santarcangelo} } @article {Bianchini2013, title = {The Monge Problem for Distance Cost in Geodesic Spaces}, journal = {Communications in Mathematical Physics}, volume = {318}, number = {3}, year = {2013}, month = {Mar}, pages = {615{\textendash}673}, abstract = {

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.

}, issn = {1432-0916}, doi = {10.1007/s00220-013-1663-8}, url = {https://doi.org/10.1007/s00220-013-1663-8}, author = {Stefano Bianchini and Fabio Cavalletti} } @article {Cavalletti2012, title = {The Monge problem in Wiener space}, journal = {Calculus of Variations and Partial Differential Equations}, volume = {45}, number = {1}, year = {2012}, month = {Sep}, pages = {101{\textendash}124}, abstract = {

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.

}, issn = {1432-0835}, doi = {10.1007/s00526-011-0452-5}, url = {https://doi.org/10.1007/s00526-011-0452-5}, author = {Fabio Cavalletti} } @article {doi:10.1137/100801433, title = {Optimal Transport with Branching Distance Costs and the Obstacle Problem}, journal = {SIAM Journal on Mathematical Analysis}, volume = {44}, number = {1}, year = {2012}, pages = {454-482}, doi = {10.1137/100801433}, url = {https://doi.org/10.1137/100801433}, author = {Fabio Cavalletti} } @conference {10.1007/978-1-4419-9554-4_10, title = {The Monge Problem in Geodesic Spaces}, booktitle = {Nonlinear Conservation Laws and Applications}, year = {2011}, pages = {217{\textendash}233}, publisher = {Springer US}, organization = {Springer US}, address = {Boston, MA}, abstract = {

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.

}, isbn = {978-1-4419-9554-4}, author = {Stefano Bianchini and Fabio Cavalletti}, editor = {Alberto Bressan and Chen, Gui-Qiang G. and Marta Lewicka and Wang, Dehua} }