On directional Whitney inequality

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.]