Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope

In this paper we make a survey of some recent developments of the theory of Sobolev spaces W1,q(X, d, m), 1 < q < ∞, in metric measure spaces (X, d, m). In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on Γ-convergence; this result extends Cheeger’s work because no Poincar´e inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of m. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.