  We study bracketing covering numbers for spaces of bounded convex functions
in the $L_p$ norms. Bracketing numbers are crucial quantities for understanding
asymptotic behavior for many statistical nonparametric estimators. Bracketing
number upper bounds in the supremum distance are known for bounded classes that
also have a fixed Lipschitz constraint. However, in most settings of interest,
the classes that arise do not include Lipschitz constraints, and so standard
techniques based on known bracketing numbers cannot be used. In this paper, we
find upper bounds for bracketing numbers of classes of convex functions without
Lipschitz constraints on arbitrary polytopes. Our results are of particular
interest in many multidimensional estimation problems based on convexity shape
constraints.
  Additionally, we show other applications of our proof methods; in particular
we define a new class of multivariate functions, the so-called $m$-monotone
functions. Such functions have been considered mathematically and statistically
in the univariate case but never in the multivariate case. We show how our
proof for convex bracketing upper bounds also applies to the $m$-monotone case.
