Convex polynomial approximation in $R^d$ with Freud weights

We show that for multivariate Freud-type weights $W_{\alpha }(x)=\exp (-\mathrm{\mid }x{\mathrm{\mid }}^{\alpha })$, $\alpha >1$, any convex function $f$ on $R^d$ satisfying $f{W}_{\alpha }\in L_p(R^d)$ if $1\le p<\mathrm{\infty }$, or ${\lim }_{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}f(x)W_{\alpha }(x)=0$ if $p=\mathrm{\infty }$, can be approximated in the weighted norm by a sequence $P_n$ of algebraic polynomials convex on $R^d$ such that $\mathrm{\parallel }(f-P_n)W_{\alpha }{\mathrm{\parallel }}_{L_p(R^d)}\to 0$ as $n\to \mathrm{\infty }$. This extends the previously known result for $d=1$ and $p=\mathrm{\infty }$ obtained by the first author to higher dimensions and integral norms using a completely different approach.