Classifying Poincar\'e Inequalities and the local geometry of RNP-Differentiability Spaces

We characterize complete RNP-differentiability spaces as those spaces which are rectifiable in terms of doubling metric measure spaces satisfying some local $(1, p)$-Poincar\'e inequalities. This gives a full characterization of spaces admitting a strong form of a differentiability structure in the sense of Cheeger, and provides a partial converse to his theorem. The proof is based on a new `thickening' construction, which can be used to enlarge subsets into spaces admitting Poincar\'e inequalities. We also introduce a new notion of quantitative connectivity which characterizes spaces satisfying local Poincar\'e inequalities. This characterization is of independent interest, and has several applications separate from differentiability spaces. We resolve a question of Tapio Rajala on the existence of Poincar\'e inequalities for the class of $MCP(K, n)$-spaces which satisfy a weak Ricci-bound. We show that deforming a geodesic metric measure space by Muckenhoupt weights preserves the property of possessing a Poincar\'e inequality. Finally, the new condition allows us to show that many classes of weak, Orlicz and non-homogeneous Poincar\'e inequalities `self-improve' to classical $(1, q)$-Poincar\'e inequalities for some $q \in [1,\infty)$, which is related to Keith's and Zhong's theorem on self-improvement of Poincar\'e inequalities.