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This paper deals with the consistency of the least squares estimator of aconvex regression function when the predictor is multidimensional. Wecharacterize and discuss the computation of such an estimator via the solutionof certain quadratic and linear programs. Mild sufficient conditions for theconsistency of this estimator and its subdifferentials in fixed and stochasticdesign regression settings are provided. We also consider a regression functionwhich is known to be convex and componentwise nonincreasing and discuss thecharacterization, computation and consistency of its least squares estimator.

Nonparametric Least Squares Estimation of a Multivariate Convex Regression Function