An intrinsic description of functions defined on a plane convex domain and having a prescribed order approximation by algebraic polynomials

Let $0<\alpha$, $0<p\le\infty$, $m$ a positive integer, let $f$ a function defined on a plane convex domain $G$. Denote by $E_m(f,L_p(G))$ the best approximation of $f$ in $L_p(G)$ by algebraic polynomials of degree $m$. A description of functions $f\in L_p(G)$ such that inequalities hold
$$ E_m(f,L_p(G))\le Cm^{-\alpha},\qquad m=1,2,\dots, $$
is given. Bibliography: 7 titles.