On directional Whitney inequality

TL;DR

This paper introduces a directional Whitney inequality on compact domains in Euclidean space, analyzing the minimal number of directions needed for the inequality to hold universally across different convex bodies and domains.

Contribution

It establishes that no universal finite set of directions works for all convex bodies, but for smooth domains, a set of d independent directions suffices, and it determines the minimal number of directions needed for various classes of domains.

Findings

No universal finite set of directions exists for all convex bodies.

For smooth domains, d independent directions suffice.

The minimal number of directions is d for certain classes of domains.

Abstract

This paper studies a new Whitney type inequality on a compact domain $\Omega\subset {\mathbb{R}}^d$ that takes the form $$\inf_{Q\in \Pi_{r-1}^d({\mathcal{E}})} \|f-Q\|_p \leq C(p,r,\Omega) \omega_{{\mathcal{E}}}^r(f,{\rm diam}(\Omega))_p,\ \ r\in {\mathbb{N}},\ \ 0<p\leq \infty,$$ where $\omega_{{\mathcal{E}}}^r(f, t)_p$ denotes the $r$-th order directional modulus of smoothness of $f\in L^p(\Omega)$ along a finite set of directions ${\mathcal{E}}\subset {\mathbb{S}^{d-1}}$ such that ${\rm span}({\mathcal{E}})={\mathbb{R}}^d$, $\Pi_{r-1}^d({\mathcal{E}}):=\{g\in C(\Omega):\ \omega^r_{\mathcal{E}} (g, {\rm diam} (\Omega))_p=0\}$. We prove that there does not exist a universal finite set of directions ${\mathcal{E}}$ for which this inequality holds on every convex body $\Omega\subset {\mathbb{R}}^d$, but for every connected $C^2$-domain $\Omega\subset {\mathbb{R}}^d$, one can choose ${\mathcal{E}}$ to be an arbitrary set of $d$ independent directions. We also study the smallest number ${\mathcal{N}}_d(\Omega)\in{\mathbb{N}}$ for which there exists a set of ${\mathcal{N}}_d(\Omega)$ directions ${\mathcal{E}}$ such that ${\rm span}({\mathcal{E}})={\mathbb{R}}^d$ and the directional Whitney inequality holds on $\Omega$ for all $r\in{\mathbb{N}}$ and $p>0$. It is proved that ${\mathcal{N}}_d(\Omega)=d$ for every connected $C^2$-domain $\Omega\subset {\mathbb{R}}^d$, for $d=2$ and every planar convex body $\Omega\subset {\mathbb{R}}^2$, and for $d\ge 3$ and every almost smooth convex body $\Omega\subset {\mathbb{R}}^d$. [See the pre-print for the complete abstract - not included here due to arXiv limitations.]