Let $$\varOmega $$Ω be a bounded closed convex set in $$\mathbb {R}^d$$Rd with nonempty interior, and let $${\mathcal C}_r(\varOmega )$$Cr(Ω) be the class of convex functions on $$\varOmega $$Ω with $$L^r$$Lr-norm bounded by 1. We obtain sharp estimates of the $$\varepsilon $$ε-entropy of $${\mathcal C}_r(\varOmega )$$Cr(Ω) under $$L^p(\varOmega )$$Lp(Ω) metrics, $$1\le p\frac{dr}{d+(d-1)r}$$p>drd+(d-1)r is attained by the closed unit ball. While a general convex body can be approximated by inscribed polytopes, the entropy rate does not carry over to the limiting body. Our results have applications to questions concerning rates of convergence of nonparametric estimators of high-dimensional shape-constrained functions.

中文翻译：

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$$\mathbb {R}^d$$ R d 上凸函数的熵

令 $$\varOmega $$Ω 是 $$\mathbb {R}^d$$Rd 内非空内部的有界闭凸集，令 $${\mathcal C}_r(\varOmega )$$Cr(Ω ) 是 $$\varOmega $$Ω 上的凸函数类，$$L^r$$Lr-范数以 1 为界。我们获得了 $$\varOmega $$ε-熵的精确估计值数学 C}_r(\varOmega )$$Cr(Ω) 在 $$L^p(\varOmega )$$Lp(Ω) 度量下，$$1\le p\frac{dr}{d+(d-1)r}$$p>drd+(d-1)r 由封闭的单位球获得。虽然一般的凸体可以用内接多胞体来近似，但熵率不会延续到极限体。我们的结果适用于有关高维形状约束函数的非参数估计量的收敛率的问题。