Alexandrov spaces with non negative curvature and displacement convexity of the entropy tensor

Abstract

On a smooth Riemannian manifold, Aishwarya, Rotem and Shenfeld characterised nonnegative sectional curvature as the matrix displacement convexity of an entropy tensor, the Lagrangian, matrix-valued refinement of Shenfeld's entropy matrix. In order to extend the entropy tensor to a finite-dimensional Alexandrov space of curvature bounded below, we construct a parallel trivialisation satisfying both the cocycle property and the second variation formula. The construction is strongly inspired by Petrunin's synthetic parallel transport. The entropy tensor defined is taken in block-diagonal form; on smooth manifolds the resulting convexity property still characterises nonnegative sectional curvature exactly. We show that the smooth equivalence persists synthetically: an Alexandrov space has nonnegative curvature if and only if its entropy tensor is matrix displacement convex.