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}},\ \ 00$. 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.]