Optimal transport and 1-Lipschitz maps

The main topic of this work is concerned with optimal transport approach to the localisation technique. The localisation technique allows to reduce a high-dimensional problem to a collection of one-dimensional problems. Significance of this approach is illustrated by its applications to Poincaré inequality, log-Sobolev inequality, isoperimetric inequality in the context of metric measure spaces satisfying the curvature-dimension condition. We investigate multi-dimensional generalisation of the technique and related conjectures of Klartag. The settlement of these conjectures would constitute a step towards a proof of the waist inequality in the setting of metric measure spaces satisfying the curvature-dimension condition. This, in turn, would help in answering the Bourgain’s hyperplane conjecture and the isoperimetric conjecture of Kannan, Lovász and Simonovits. We provide a partial affirmative answer to the conjecture that, whenever Euclidean space equipped with a measure satisfies the curvature-dimension condition, then any vector-valued 1-Lipschitz map induces a partition of the domain such that the leaves equipped with the conditional measures satisfy the curvature-dimension condition with the same parameters as the initial space did. Our approach is based on the ideas of Sudakov and their further instances in works of Caffarelli, Feldman, McCann and of Klartag. We provide a counterexample to another conjecture of Klartag that the conditional measures, with respect to a partition induced by a certain 1-Lipschitz map, satisfy so-called mass balance condition. We develop a theory of optimal transport of vector measures and employ it to provide a sufficient condition for the mass balance condition to hold true. One way of proving the mass balance condition in one-dimensional setting is via approximation of Lipshitz functions. This leads us to a study of continuity properties of extensions of vector-valued Lipschitz maps. We provide a strengthening of the classical Kirszbraun theorem, which shows a discrepancy between one-dimensional and multi-dimensional cases. We identify a sharp rate of continuity of extensions of 1-Lipschitz maps. Recently, several variants of the optimal transport problem have been studied. Martingale optimal transport is a one of the greatest importance. Interestingly, also in this case a localisation result has been established. Therefore, with help of a novel version of Strassen’s theorem, we investigate the duality theorems in this problem. We provide a reformulation of the optimal transport in two-marginal, in multi-marginal and in martingale setting. We reprove the Kantorovich duality formulae. As an application we characterise the uniformly smooth and the uniformly convex functions. We develop a divergence formulation of optimal transport of vector measures and use it to provide a novel proof of the representation formula for polar cone to monotone maps. We find several generalisations of this result. A tool that we use is the matrix Hölder’s inequality, for which we characterise the equality cases.