  In this work, we consider a multivariate regression model with one-sided
errors. We assume for the regression function to lie in a general H\"{o}lder
class and estimate it via a nonparametric local polynomial approach that
consists of minimization of the local integral of a polynomial approximation
lying above the data points. While the consideration of multivariate covariates
offers an undeniable opportunity from an application-oriented standpoint, it
requires a new method of proof to replace the established ones for the
univariate case. The main purpose of this paper is to show the uniform
consistency and to provide the rates of convergence of the considered
nonparametric estimator for both multivariate random covariates and
multivariate deterministic design points. To demonstrate the performance of the
estimators, the small sample behavior is investigated in a simulation study in
dimension two and three.
