Riemannian manifolds of L(p)-positive Ricci curvature

We study manifolds in which the amount of Ricci curvature below a positive constant is small in an Lp sense. In particular, we present optimal generalizations of Myers' diameter bound, volume comparison, and mean curvature comparison. Since these are some of the main tools in positive Ricci curvature, we can conclude that many standard results will carry over to our situation.