Separating disconnected quadratic level sets by other quadratic level sets

Given a quadratic function f(x)=x^TAx+2a^Tx+a₀ and its level sets, including {x∈Rⁿ∣f(x)<0}, {x∈Rⁿ∣f(x)≤0}, {x∈Rⁿ∣f(x)=0}, {x∈Rⁿ∣f(x)≥0}, {x∈Rⁿ∣f(x)>0}, we derive conditions on the coefficients (A,a,a₀) 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∈Rⁿ∣g(x)=x^TBx+2b^Tx+b₀=0} such that L^-⊂{x∈Rⁿ∣g(x)<0} and L⁺⊂{x∈Rⁿ∣g(x)>0}. It has been shown that the particular case for {x∈Rⁿ∣g(x)=0} to separate {x∈Rⁿ∣f(x)<0} is equivalent to the S-lemma with equality, and thus answers the strong duality for min{f(x)∣g(x)=0}. In addition, the case when {x∈Rⁿ∣g(x)=0} separates {x∈Rⁿ∣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∈Rⁿ∣g(x)=0} to separate {x∈Rⁿ∣f(x)⋆0} where ⋆∈{<,≤,=,≥,>}, which is expected to provide a new tool in solving quadratic optimization problems.