At the close of the 1980s, the independent contributions of Yann Brenier, Mike Cullen and
John Mather launched a revolution in the venerable field of optimal transport founded by G …

We introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is
more restrictive than the curvature bound Curv\left(M,d,m\right)>K (introduced in Sturm KT …

We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD (K, N)
metric measure spaces; regularity is understood with respect to a newly defined quasi …

Abstract We extend Cordero-Erausquin et al.'s Riemannian Borell–Brascamp–Lieb
inequality to Finsler manifolds. Among applications, we establish the equivalence between …

Abstract 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 …

We prove local Poincaré inequalities under various curvature-dimension conditions which
are stable under the measured Gromov–Hausdorff convergence. The first class of spaces …

We survey recent advances in analysis and geometry, where first order differential analysis
has been extended beyond its classical smooth settings. Such studies have applications to …

The main result of this article states that the (K, N)-cone over some metric measure space
satisfies the reduced Riemannian curvature-dimension condition RCD⁎(KN, N+ 1) if and …

This paper studies the structure and stability of boundaries in noncollapsed RCD (K, N)
spaces, that is, metric-measure spaces (X, d, HN) with Ricci curvature bounded below. Our …

We give sufficient conditions for a measured length space (X, d, ν) to admit local and global
Poincaré inequalities, along with a Sobolev inequality. We first introduce a condition DM on …