Open AccessEntropy of Convex Functions on $$\mathbb {R}^d$$ R d
Author(s) -
Fuchang Gao,
Jon A. Wellner
Publication year - 2017
Publication title -
constructive approximation
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.921
H-Index - 51
eISSN - 1432-0940
pISSN - 0176-4276
DOI - 10.1007/s00365-017-9387-1
Subject(s) - combinatorics , mathematics , polytope , upper and lower bounds , bounded function , convex body , regular polygon , unit sphere , convex function , unit cube , mathematical analysis , geometry , convex hull
Let Ω be a bounded closed convex set in ℝ d with non-empty interior, and let 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 r (Ω) under L p (Ω) metrics, 1 ≤ p < r ≤ ∞. In particular, the results imply that the universal lower bound ε - d /2 is also an upper bound for all d -polytopes, and the universal upper bound of [Formula: see text] for [Formula: see text] 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.