On the convexity for the range set of two quadratic functions

Given \begin{document}$ n\times n $\end{document} symmetric matrices \begin{document}$ A $\end{document} and \begin{document}$ B, $\end{document} Dines in 1941 proved that the joint range set \begin{document}$ \{(x^TAx, x^TBx)|\; x\in\mathbb{R}^n\} $\end{document} is always convex. Our paper is concerned with non-homogeneous extension of the Dines theorem for the range set \begin{document}$ \mathbf{R}(f, g) = \{\left(f(x), g(x)\right)|\; x \in \mathbb{R}^n \}, $\end{document}\begin{document}$ f(x) = x^T A x + 2a^T x + a_0 $\end{document} and \begin{document}$ g(x) = x^T B x + 2b^T x + b_0. $\end{document} We show that \begin{document}$ \mathbf{R}(f, g) $\end{document} is convex if, and only if, any pair of level sets, \begin{document}$ \{x\in\mathbb{R}^n|f(x) = \alpha\} $\end{document} and \begin{document}$ \{x\in\mathbb{R}^n|g(x) = \beta\} $\end{document} , do not separate each other. With the novel geometric concept about separation, we provide a polynomial-time procedure to practically check whether a given \begin{document}$ \mathbf{R}(f, g) $\end{document} is convex or not.