(K, N)-Convexity and the Curvature-Dimension Condition for Negative N

We extend the range of N to negative values in the (K, N)-convexity (in the sense of Erbar-Kuwada-Sturm), the weighted Ricci curvature RicN and the curvaturedimension condition CD(K, N). We generalize a number of results in the case of N > 0 to this setting, including Bochner's inequality, the Brunn-Minkowski inequality and the equivalence between Ric(N) >= K and CD(K, N). We also show an expansion bound for gradient flows of Lipschitz (K, N)-convex functions.