Products of functions with bounded ${\rm Hess}^+$ complement

Abstract

We denote by ${\rm Hess}^+$ the set of all points $p\in\mathbb{R}^n$ such that the Hessian matrix $H_p(f)$ of the $C^2$-smooth function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ is positive definite. In this paper we provide a class of norm-coercive polynomial functions with large ${\rm Hess}^+$ regions, as their ${\rm Hess}^+$ complements happen to be bounded. A detailed analysis concerning the ${\rm Hess}^+$ region of a particular polynomial function along with some basic properties of its level curves, such as regularity, connectedness and convexity, is also provided. For such functions we also prove several properties, such as connectedness and convexity, of their level sets for sufficiently large levels. Apart from the mentioned source of such examples we provide some sufficient conditions on two functions $f,g:\mathbb{R}^2\longrightarrow\mathbb{R}$ with bounded ${\rm Hess}^+$ complements whose product $fg$ keeps having bounded ${\rm Hess}^+$ complement as well.