We prove a new estimate on manifolds with a lower Ricci bound which asserts that the geometry of balls centered on a minimizing geodesic can change in at most a Hölder continuous way along the geodesic. We give examples that show that the Hölder exponent, along with essentially all the other consequences that follow from this estimate, are sharp.
Among the applications is that the regular set is convex for any noncollapsed limit of Einstein metrics. In the general case of a potentially collapsed limit of manifolds with just a lower Ricci curvature bound we show that the regular set is weakly convex and a.e. convex. We also show two conjectures of Cheeger-Colding. One of these asserts that the isometry group of any, even collapsed, limit of manifolds with a uniform lower Ricci curvature bound is a Lie group. The other asserts that the dimension of any limit space is the same everywhere.