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We extend the range of $N$ to negative values in the $(K,N)$-convexity (inthe sense of Erbar--Kuwada--Sturm), the weighted Ricci curvature $Ric_N$ andthe curvature-dimension condition $CD(K,N)$. We generalize a number of resultsin the case of $N>0$ to this setting, including Bochner's inequality, theBrunn--Minkowski inequality and the equivalence between $Ric_N \ge K$ and$CD(K,N)$. We also show an expansion bound for gradient flows of Lipschitz$(K,N)$-convex functions.

$(K,N)$-convexity and the curvature-dimension condition for negative $N$