Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications

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 Hlder continuous way along the geodesic. We give examples that show that the Hlder exponent, along with essentially all the other consequences that follow from this estimate, are sharp.