Exponential convexifying of polynomials

Abstract

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 $N\ge 1$, such that for any $\xi \in {\mathbb{R}}^n$ the function ${\phi }_N(x)=e^{N|x-\xi {|}^2}f(x)$ 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 ${\Phi }_N(x)=e^{e^{N|x{|}^2}}f(x)$ 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.

19 pages