Bracketing Numbers of Convex Functions on Polytopes

We study bracketing numbers for spaces of bounded convex functions in the $L_p$ norms. We impose no Lipschitz constraint. Previous results gave bounds when the domain of the functions is a hyperrectangle. We extend these results to the case wherein the domain is a polytope. Bracketing numbers are crucial quantities for understanding asymptotic behavior for many statistical nonparametric estimators. Our results are of interest in particular in many multidimensional estimation problems based on convexity shape constraints.