  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.]
