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

TL;DR

This paper demonstrates that the local CD(K,N) condition does not necessarily imply the global CD(K,N) condition in certain metric measure spaces, challenging assumptions about their equivalence.

Contribution

It constructs explicit examples of spaces satisfying local CD(0,4) but failing global CD(K,N), showing the failure of the local-to-global property in these spaces.

Findings

Existence of compact spaces with local CD(0,4) but not global CD(K,N)

Construction of non-convex subsets of R^2 with specific metric properties

Extension to non-compact spaces satisfying local but not global CD conditions

Abstract

Given any K and N 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 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.