For a positive polynomial $fin mathbb{R}[x_1,ldots,x_n]$ we give necessary and sufficient conditions to existence of an exponent $Ninmathbb{N}$ such that $(1+|x|^2)^Nf(x)$ is a convex function, where $|x|^2={x_1^2+cdots+x_n^2}$. Next we show that if $finmathbb{R}[x_1,ldots,x_n]$ is strictly positive on a closed convex basic semialgebraic set $X={xinmathbb{R}^n:g_1(x)ge 0,ldots,g_r (x)ge 0}$, where $g_1,ldots,g_rinmathbb{R}[x_1,ldots,x_n]$ are concave polynomials, then $f$ can be approximated (in the $l_1$ norm) by polynomials of the quadratic module $Q(g_1,ldots,g_r)$. In the case $X=mathbb{R}^n$ the approximation is uniform on compact sets. Joint work with S. Spodzieja.