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.]
Categories: math.NA cs.NA math.CA