  This paper deals with the consistency of the least squares estimator of a
convex regression function when the predictor is multidimensional. We
characterize and discuss the computation of such an estimator via the solution
of certain quadratic and linear programs. Mild sufficient conditions for the
consistency of this estimator and its subdifferentials in fixed and stochastic
design regression settings are provided. We also consider a regression function
which is known to be convex and componentwise nonincreasing and discuss the
characterization, computation and consistency of its least squares estimator.
