具有有界${\rm Hess}^+$补的函数的乘积
Products of functions with bounded ${\rm Hess}^+$ complement
摘要：
我们用${\rm Hess}^+$表示所有点$p\in\mathbb{R}^n$的集合，使得$C^2$ -光滑函数$f:\mathbb{R}^n\longrightarrow\mathbb{R}$的Hessian矩阵$H_p(f)$是正定的。本文给出了一类具有大${\rm Hess}^+$区域的范数强制多项式函数，因为它们的${\rm Hess}^+$补函数是有界的.详细分析了一个特定多项式函数的${\rm Hess}^+$区域及其水平曲线的一些基本性质，如正则性、连通性和凸性。对于这样的功能，我们也证明了几个属性，如连通性和凸性，他们的水平集足够大的水平。除了上述例子的来源外，我们还给出了两个函数$f,g:\mathbb{R}^2\longrightarrow\mathbb{R}$具有有界${\rm Hess}^+$补且其乘积$fg$也保持有界${\rm Hess}^+$补的充分条件.
英文摘要：
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.
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