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 $W^{1,q}(X,\text{sfd},\text{mm})$, $1<q<\mathrm{\infty }$, in metric measure spaces $(X,\text{sfd},\text{mm})$. In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on $\mathrm{\Gamma }$-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 $\text{mm}$. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.