On Separation of level sets for a pair of quadratic functions

Given a quadratic function $f(x)=x^TAx+2a^Tx+a_0,$ it is possible that its level set $\{x\in\mathbb{R}^n: f(x)=0\}$ has two connected components and thus can be separated by the level set $\{x\in\mathbb{R}^n: g(x)=0\}$ of another quadratic function $g(x)=x^TBx+2b^Tx+b_0.$ It turns out that the separation property of such kind has great implication in quadratic optimization problems and thus deserves careful studies. In this paper, we characterize the separation property analytically by necessary and sufficient conditions as a new tool to solving optimization problems.