Shape of Alexandrov spaces with positive Ricci curvature

Abstract

Under the definition of Ricci curvature bounded below for Alexandrov spaces introduced by Zhang-Zhu, we extend a result by Colding that an n−dimensional manifold with Ricci curvature greater or equal to n − 1 and volume close to that of the unit n−sphere is close (in the Gromov-Hausdorff distance) to the sphere, from the case of Riemannian manifolds to the case of Alexandrov spaces, with an additional assumption, roughly speaking, that the rough volume of the set of "short" cut points is small, following the basic idea in the Riemannian case with necessary modifications because of the only almost everywhere second differentiability of distance functions.