Separating disconnected quadratic level sets by other quadratic level sets

Given a quadratic function $$f(x) = x^T A x + 2 a^T x + a_0$$ and its level sets, including $$\{x\in \mathbb {R}^n \mid f(x) < 0\}$$ , $$\{x\in \mathbb {R}^n \mid f(x) \le 0\}$$ , $$\{x\in \mathbb {R}^n \mid f(x) = 0\}$$ , $$\{x\in \mathbb {R}^n \mid f(x) \ge 0\}$$ , $$\{x\in \mathbb {R}^n \mid f(x) > 0\},$$ we derive conditions on the coefficients $$(A, a, a_0)$$ for each of the five level sets to be disconnected with exactly two connected components, say $$L^{-}$$ and $$L^{+}$$ . Once so, we are interested in, when the two connected components can be separated by another quadratic level set $$\{x\in \mathbb {R}^n \mid g(x) = x^T B x + 2b^T x + b_0 = 0\}$$ such that $$L^{-} \subset \{x\in \mathbb {R}^n \mid g(x) < 0\}$$ and $$L^{+} \subset \{x\in \mathbb {R}^n \mid g(x) > 0\}$$ . It has been shown that the particular case for $$\{x\in \mathbb {R}^n \mid g(x) = 0\}$$ to separate $$\{x\in \mathbb {R}^n \mid f(x) < 0\}$$ is equivalent to the $$\mathcal {S}$$ -lemma with equality, and thus answers the strong duality for $$\min \{f(x) \mid g(x) = 0\}.$$ In addition, the case when $$\{x\in \mathbb {R}^n \mid g(x) = 0\}$$ separates $$\{x\in \mathbb {R}^n \mid f(x) = 0\}$$ is closely related to the convexity of the joint range for a pair of quadratic functions (f, g). This paper completes the characterization for all five cases of separation, namely, for $$\{x\in \mathbb {R}^n \mid g(x) = 0\}$$ to separate $$\{x\in \mathbb {R}^n \mid f(x) \star 0\}$$ where $$\star \in \{<, \le , =, \ge , >\}$$ , which is expected to provide a new tool in solving quadratic optimization problems.