Entropy of convex functions on R^d

2015/02/05 by Fuchang Gao, Jon A. Wellner, Gao, Fuchang +1
Mathematics · Nursing · #41A46. Secondary: 52A27 #52B11 #52C17 #FOS: Mathematics #Fatty Acid Research and Health #Mathematical Approximation and Integration #Point processes and geometric inequalities #Primary: 52A41 #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.1502.01752

openalex publication_date 2015/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ω be a bounded closed convex set in \mathbb R^d with non-empty interior, and let \cal C_r(Ω) be the class of convex functions on Ω with L^r-norm bounded by 1. We obtain sharp estimates of the ε-entropy of \cal C_r(Ω) under L^p(Ω) metrics, 1≤ p(dr)/(d+(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.

Citations

• Covering numbers of L_p-balls of convex sets and functions

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