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.

我们研究了包围凸函数空间中的包围数 ${\mathrm{大号}}_p$规范。括号中的数字对于理解许多统计非参数估计量的渐近行为至关重要。对于有固定Lipschitz约束的有界类，已知最高距离中的包围数上限。但是，在大多数感兴趣的设置中，出现的类不包括Lipschitz约束，因此无法使用基于已知包围号的标准技术。在本文中，我们找到了在任意多边形上没有Lipschitz约束的凸函数类的包围次数的上限。我们的结果在许多基于凸形状约束的多维估计问题中特别有意义。

另外，我们展示了证明方法的其他应用；特别是我们定义了一类新的多元函数，即所谓的$米$-单调功能。在单变量情况下已经从数学和统计角度考虑了此类函数，但在多变量情况下从未考虑过。我们展示了凸括号上界的证明也适用于$米$-单调的情况。