Abstract:
Let $M\subseteq\mathbf R^m$ be a compact convex body, and $O$ the center of gravity of $M$. For a convex function $f\colon M\to\mathbf R$ let
$$
\omega(f,\delta,M)=\sup_{\substack{x,y\in M\\|x-y|_M\leqslant\delta}}|f(x)-f(y)|\qquad(\delta\geqslant0),
$$
where $|x|_M=\min\{\mu\geqslant0:x\in\mu(M-O)\}$, and let $M_1\subseteq\mathbf R^m$ be a convex body, $M\subseteq M_1$, and $\varkappa=\min\{\mu\geqslant1:M_1\subseteq\mu M\}$, $\mu M$ being a homothety of $M$ with respect to $O$. Then for $n\geqslant0$ there exists an algebraic polynomial
$$
p_n(x)=\sum_{i_1+\dots+i_m\leqslant n}a_{i_1,\dots,i_m}x^{i_1}_1\cdots x^{i_m}_m
$$
that is convex on $M_1$ and such that
$$
\|f-p_n\|_{C(M)}\leqslant\varkappa A_m\omega\biggl(f,\frac1{n+1},M\biggr).
$$
Fulltext:
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