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We demonstrate the necessity of a Poincar\'e type inequality for those metricmeasure spaces that satisfy Cheeger's generalization of Rademacher's theoremfor all Lipschitz functions taking values in a Banach space with theRadon-Nikodym property. This is done by showing the existence of a richstructure of curve fragments that connect near by points, similar in nature toSemmes's pencil of curves for the standard Poincar\'e inequality. Usingtechniques similar to Cheeger-Kleiner, we show that our conditions are alsosufficient.  We also develop another characterization of "RNP Lipschitz differentiabilityspaces" by connecting points by curves that form a rich structure of partialderivatives that were first discussed in work by the first author.

Differentiability and Poincar\'e-type inequalities in metric measure spaces