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

TL;DR

This paper studies polynomial functions with large regions where the Hessian is positive definite, analyzing their level sets and providing conditions under which the product of two such functions also has a bounded Hessian positive region.

Contribution

It introduces a class of polynomial functions with large Hess^+ regions and analyzes their level sets, also establishing conditions for products of functions to maintain bounded Hess^+ complements.

Findings

Constructed polynomial examples with large Hess^+ regions

Analyzed properties of level sets such as convexity and connectedness

Provided conditions for products of functions to have bounded Hess^+ complements

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.