Whitney-type estimates for convex functions

TL;DR

The authors address Whitney-type inequalities for convex function approximation on convex bodies, introducing three constants $w_m(K)$, $\breve w_m(K)$, and $\breve{\breve w}_m(K)$ to capture unconstrained, convexity-restricted, and convexity-preserving approximation. They show that for linear approximants ($m=2$) convexity constraints alter the constants significantly, with $\breve w_{2,n}$ scaling like $\tfrac{1}{4}\log_2 n$ and exact values $\breve w_2(K)=\tfrac{1}{2}$ in the centrally symmetric case, while the unconstrained symmetric constants diverge; for higher-degree approximants ($m\ge 3$) the convexity constraint does not improve the constants, i.e., $\breve w_m(K)=w_m(K)$. In the convexity-preserving setting, the landscape changes dramatically: $m\ge 4$ yields infinite constants, while $m=3$ admits distance-dependent finite bounds via the Banach–Mazur distance, with explicit results for balls and symmetric domains and a discussion of non-uniqueness of best approximants. Overall, the work provides dimension- and geometry-aware bounds that illuminate how convexity and shape-preserving requirements interact with multivariate polynomial approximation.

Abstract

We study Whitney-type estimates for approximation of convex functions in the uniform norm on various convex multivariate domains while paying a particular attention to the dependence of the involved constants on the dimension and the geometry of the domain.

Theorems & Definitions (11)

• Theorem 1.1: BrKa*Th. 3.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Proposition 1.8
• Proposition 2.1: BrKa*Prop. 3.2
• Corollary 2.2