*Preprint*

**Inserted:** 29 may 2013

**Last Updated:** 29 may 2013

**Year:** 2013

**Abstract:**

Given any $K \in \mathbb{R}$ and $N \in [1,\infty]$ we show that there exists a compact geodesic metric measure space satisfying locally the $CD(0,4)$ condition but failing $CD(K,N)$ globally. The space with this property is a suitable non convex subset of $\mathbb{R}^2$ equipped with the $l^\infty$-norm and the Lebesgue measure. Combining many such spaces gives a (non compact) complete geodesic metric measure space satisfying $CD(0,4)$ locally but failing $CD(K,N)$ globally for every $K$ and $N$.

**Keywords:**
entropy, Ricci curvature, metric measure spaces, branching geodesics

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