A multidimensional analog of a theorem of Whitney

The following theorem is proved:
Theorem. {\it Let $f\in L_p(\Omega)$, where $\Omega$ is a convex domain in $R^n$. Then
$$ \inf_l\|f-l\| _{L_p(\Omega)}\leqslant w\sup_h\|\Delta_h^kf\|, $$
where the $\inf$ on the left is taken over all degree $k-1$ polynomials, and the $L_p$ norm on the right is taken over the set in which the $k$th difference $\Delta_h^kf$ is defined. The constant $w$ depends only on $k,n$, and the ratio of the diameter of $\Omega$ to its width}.
H. Whitney proved this theorem in the case $p=\infty$ and $\Omega=[0,1]$. As a corollary, it is proved that the $k$-modulus of continuity dominates any “deviation”, constructed with the help of a measure with compact support, orthogonal to polynomials of degree $k-1$.
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