  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.
