On directional Whitney inequality

Abstract This paper studies a new Whitney type inequality on a compact domain $\Omega \subset {\mathbb R}^d$ that takes the form $$ \begin{align*} \inf_{Q\in \Pi_{r-1}^d(\mathcal{E})} \|f-Q\|_p \leq C(p,r,\Omega) \omega_{\mathcal{E}}^r(f,\mathrm{diam}(\Omega))_p,\ \ r\in {\mathbb N},\ \ 0&lt;p\leq \infty, \end{align*} $$ 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 $\mathrm {span}(\mathcal {E})={\mathbb R}^d$ , $\Pi _{r-1}^d(\mathcal {E}):=\{g\in C(\Omega ):\ \omega ^r_{\mathcal {E}} (g, \mathrm {diam} (\Omega ))_p=0\}$ . 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 $\mathrm {span}(\mathcal {E})={\mathbb R}^d$ and the directional Whitney inequality holds on $\Omega $ for all $r\in {\mathbb N}$ and $p&gt;0$ . 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 $\Omega \subset {\mathbb R}^d$ . For $d\ge 3$ and a more general convex body $\Omega \subset {\mathbb R}^d$ , we connect $\mathcal {N}_d(\Omega )$ with a problem in convex geometry on the X-ray number of $\Omega $ , proving that if $\Omega $ is X-rayed by a finite set of directions $\mathcal {E}\subset \mathbb {S}^{d-1}$ , then $\mathcal {E}$ </jats:tex-math

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