In this paper, we survey and generalize results on the R-convex hull of the points of a random sample. We provide explicit convergence rates for the R-convex hull and its natural plug-ins, as estimators of the support S , its volume, perimeter, and level-sets of a multidimensional density , when the underlying sequence is strictly stationary and admits some kind of weak dependence condition. We obtain convergence rates that exceed those obtained in the i.i.d. case, depending on the dependence condition.