Abstract. We prove that if (X, d, m) is a metric measure space with m(X) = 1 having (in a synthetic sense) Ricci curvature bounded from below by K > 0 and dimension bounded above by N ∈ [1, ∞), then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by E. Milman for any K ∈ R, N ≥ 1 and upper diameter bounds) holds, i.e. the isoperimetric profile function of (X, d, m) is bounded from below by the isoperimetric profile of the model space. Moreover, if equality is attained for some volume v ∈ (0, 1) and K is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions. Finally we also establish the almost rigidity: if the equality is almost attained for some volume v ∈ (0, 1) and K is strictly positive, then the space must be mGH close to a spherical suspension. To our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying Ricci curvature lower bounds. Examples of spaces fitting our assumptions include measured Gromov-Hausdorff limits of Riemannian manifolds satisfying Ricci curvature lower bounds, Alexandrov spaces with curvature bounded from below, Finsler manifolds endowed with a strongly convex norm and satisfying Ricci curvature lower bounds; the result seems new even in these celebrated classes of spaces.

The Lott-Sturm-Villani Curvature-Dimension condition provides a synthetic notion for a metric-measure space to have Ricci-curvature bounded from below and dimension bounded from above. We prove that it is enough to verify this condition locally: an essentially non-branching metric-measure space (X, d, m) (so that (supp(m), d) is a length-space and m(X) < ∞) verifying the local Curvature-Dimension condition CD loc (K, N ) with parameters K ∈ R and N ∈ (1, ∞), also verifies the global Curvature-Dimension condition CD(K, N ), meaning that the Curvature-Dimension condition enjoys the globalization (or local-to-global) property. The main new ingredients of our proof are an explicit change-of-variables formula for densities of Wasserstein geodesics depending on a second-order derivative of an associated Kantorovich potential; a surprising third-order bound on the latter Kantorovich potential, which holds in complete generality on any proper geodesic space; and a certain rigidity property of the change-of-variables formula, allowing us to bootstrap the a-priori available regularity. The change-of-variables formula is obtained via a new synthetic notion of Curvature-Dimension we dub CD 1 (K, N ). Contents

We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and d L is a geodesic Borel distance which makes (X, d L ) 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 1dimensional Hausdorff distance induced by d L . 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 d L -cyclical monotonicity is not sufficient for optimality.

We prove that for non-branching metric measure spaces the local curvature condition CD loc (K, N ) implies the global version of MCP(K, N ). The curvature condition CD(K, N ) introduced by the second author and also studied by Lott & Villani is the generalization to metric measure space of lower bounds on Ricci curvature together with upper bounds on the dimension. This paper is the following step of [1] where it is shown that CD loc (K, N ) is equivalent to a global condition CD * (K, N ), slightly weaker than the usual CD(K, N ). It is worth pointing out that our result implies sharp Bishop-Gromov volume growth inequality and sharp Poincaré inequality.

The Lott–Sturm–Villani Curvature-Dimension condition provides a synthetic notion for a metric-measure space to have Ricci-curvature bounded from below and dimension bounded from above. We prove that it is enough to verify this condition locally: an essentially non-branching metric-measure space $$(X,\mathsf {d},{\mathfrak {m}})$$ ( X , d , m ) (so that $$(\text {supp}({\mathfrak {m}}),\mathsf {d})$$ ( supp ( m ) , d ) is a length-space and $${\mathfrak {m}}(X) < \infty $$ m ( X ) < ∞ ) verifying the local Curvature-Dimension condition $${\mathsf {CD}}_{loc}(K,N)$$ CD loc ( K , N ) with parameters $$K \in {\mathbb {R}}$$ K ∈ R and $$N \in (1,\infty )$$ N ∈ ( 1 , ∞ ) , also verifies the global Curvature-Dimension condition $${\mathsf {CD}}(K,N)$$ CD ( K , N ) . In other words, the Curvature-Dimension condition enjoys the globalization (or local-to-global) property, answering a question which had remained open since the beginning of the theory. For the proof, we establish an equivalence between $$L^1$$ L 1 - and $$L^2$$ L 2 -optimal-transport–based interpolation. The challenge is not merely a technical one, and several new conceptual ingredients which are of independent interest are developed: an explicit change-of-variables formula for densities of Wasserstein geodesics depending on a second-order temporal derivative of associated Kantorovich potentials; a surprising third-order theory for the latter Kantorovich potentials, which holds in complete generality on any proper geodesic space; and a certain rigidity property of the change-of-variables formula, allowing us to bootstrap the a-priori available regularity. As a consequence, numerous variants of the Curvature-Dimension condition proposed by various authors throughout the years are shown to, in fact, all be equivalent in the above setting, thereby unifying the theory.