Polynomial and rational approximation of functions of several variables with convex derivatives in the $L_p$-metric $(0<p\leqslant\infty)$

Let $\operatorname{Conv}_n^{(l)}(\mathscr G)$ be the set of all functions $f$ such that for every $n$-dimensional unit vector $\mathbf e$ the $l$th derivative in the direction of $\mathbf e$, $D^{(l)}(\mathbf e)f$, is continuous on a convex bounded domain $\mathscr G\subset\mathbf R^n$ $(n\geqslant 2)$ and convex (upwards or downwards) on the nonempty intersection of every line $L\subset\mathbf R^n$ with the domain $\mathscr G$, and let $M^{(l)}(f,\mathscr G)\colon=\sup\{\|D^{(l)}(\mathbf e)f\|_{C(\mathscr G)}\colon\mathbf e\in \mathbf R^n$, $\|\mathbf e\|=1\}<\infty$. Sharp, in the sense of order of smallness, estimates of best simultaneous polynomial approximations of the functions $f\in\operatorname{Conv}_n^{(l)}(\mathscr G)$ for which $D^{(l)}(\mathbf e)f\in\operatorname{Lip}_K1$ for every $\mathbf e$, and their derivatives in the metrics of $L_p(\mathscr G)$ $(0<p\leqslant\infty)$ are obtained. It is proved that the corresponding parts of these estimates are preserved for best rational approximations, on any $n$-dimensional parallelepiped $Q$, of functions $f\in\operatorname{Conv}_n^{(l)}(Q)$ in the metrics of $L_p(Q)$ $(0<p<\infty)$ and it is shown that they are sharp in the sense of order of smallness for $0<p\leqslant 1$.