  Dinh D\~ung and T. Ullrich have proven a multivariate Whitney's theorem for
the local anisotropic polynomial approximation in $L_p(Q)$ for $1 \le p \le
\infty$, where $Q$ is a $d$-parallelepiped in $\RR^d$ with sides parallel to
the coordinate axes. They considered the error of best approximation of a
function $f$ by algebraic polynomials of fixed degree at most $r_i - 1$ in
variable $x_i,\ i=1,...,d$. The convergence rate of the approximation error
when the size of $Q$ going to 0 is characterized by a so-called total mixed
modulus of smoothness. The method of proof used by these authors is not
suitable to the case $0 <p<1$. In the present paper, by a different method we
proved this theorem for $0< p \le \infty$.
