  Let $X\subset\mathbb{R}^n$ be a convex closed and semialgebraic set and let
$f$ be a polynomial positive on $X$. We prove that there exists an exponent
$N\geq 1$, such that for any $\xi\in\mathbb{R}^n$ the function
$\varphi_N(x)=e^{N|x-\xi|^2}f(x)$ is strongly convex on $X$. When $X$ is
unbounded we have to assume also that the leading form of $f$ is positive in
$\mathbb{R}^n\setminus\{0\}$. We obtain strong convexity of
$\varPhi_N(x)=e^{e^{N|x|^2}}f(x)$ on possibly unbounded $X$, provided $N$ is
sufficiently large, assuming only that $f$ is positive on $X$. We apply these
results for searching critical points of polynomials on convex closed
semialgebraic sets.
