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Given any K and N we show that there exists a compact geodesic metric measurespace 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 R^2 equippedwith the l^\infty-norm and the Lebesgue measure. Combining many such spacesgives a (non compact) complete geodesic metric measure space satisfying CD(0,4)locally but failing CD(K,N) globally for every K and N.

Failure of the local-to-global property for CD(K,N) spaces