On directional Whitney inequality

Abstract This paper studies a new Whitney type inequality on 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 N},\ 0<p\leq \infty, \end{align*} where $\omega _{\mathcal {E}}^r(f, t)_p$ denotes r th order directional modulus of smoothness $f\in L^p(\Omega )$ along finite set directions $\mathcal {E}\subset \mathbb {S}^{d-1}$ such $\mathrm {span}(\mathcal {E})={\mathbb , $\Pi _{r-1}^d(\mathcal {E}):=\{g\in C(\Omega ):\ \omega ^r_{\mathcal {E}} (g, \mathrm {diam} (\Omega ))_p=0\}$ . We prove there does not exist universal {E}$ for which this holds every convex body but connected $C^2$ -domain one can choose to be an arbitrary d independent directions. also study smallest number {N}_d(\Omega )\in N}$ exists and $ all $r\in $p>0$ It is proved )=d$ $d=2$ planar R}^2$ $d\ge 3$ almost smooth For more general we connect with problem in geometry X-ray proving if X-rayed by then admits $0<p\leq \infty Such connection allows us deduce certain quantitative estimate A slight modification proof usual literature yields each containing than $(c d)^{d-1}$ In paper, develop simpler method general, possibly nonconvex domains requiring significantly fewer moduli.