Geometric properties for level sets of quadratic functions

In this paper, we study some fundamental geometrical properties related to the $${\mathcal {S}}$$ S -procedure. Given a pair of quadratic functions (g, f), it asks when “ $$g(x)=0 \Longrightarrow ~ f(x)\ge 0$$ g ( x ) = 0 ⟹ f ( x ) ≥ 0 ” can imply “( $$\exists \lambda \in {\mathbb {R}}$$ ∃ λ ∈ R ) ( $$\forall x\in {\mathbb {R}}^n$$ ∀ x ∈ R n ) $$f(x) + \lambda g(x)\ge 0.$$ f ( x ) + λ g ( x ) ≥ 0 . ” Although the question has been answered by Xia et al. (Math Program 156:513–547, 2016), we propose a neat geometric proof for it (see Theorem 2): the $${\mathcal {S}}$$ S -procedure holds when, and only when, the level set $$\{g=0\}$$ { g = 0 } cannot separate the sublevel set $$\{f