This paper studies a new Whitney type inequality on a compact domain
$\Omega \subset {\mathbb R}^d$
that takes the form
$$ \begin{align*} \inf_{Q\in \Pi_{r-1}^d(\mathcal{E})} \|f-Q\|_p \leq C(p,r,\Omega) \omega_{\mathcal{E}}^r(f,\mathrm{diam}(\Omega))_p,\ \ r\in {\mathbb N},\ \ 0<p\leq \infty, \end{align*} $$
where
$\omega _{\mathcal {E}}^r(f, t)_p$
denotes the
rth 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
$p>0$
. 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
$\mathcal {E}$
admits the directional Whitney inequality on
$\Omega $
for all
$r\in {\mathbb N}$
and
$0<p\leq \infty $
. Such a connection allows us to deduce certain quantitative estimate of
$\mathcal {N}_d(\Omega )$
for
$d\ge 3$
.
A slight modification of the proof of the usual Whitney inequality in literature also yields a directional Whitney inequality on each convex body
$\Omega \subset {\mathbb R}^d$
, but with the set
$\mathcal {E}$
containing more than
$(c d)^{d-1}$
directions. In this paper, we develop a new and simpler method to prove the directional Whitney inequality on more general, possibly nonconvex domains requiring significantly fewer directions in the directional moduli.