ON THE CONVEXITY FOR THE RANGE SET OF TWO QUADRATIC FUNCTIONS

Given n × n symmetric matrices A and B, Dines in 1941 proved that the joint range set {(x^TAx, x^TBx) | x ∈ ℝⁿ} is always convex. Our paper is concerned with non-homogeneous extension of the Dines theorem for the range set R(f, g) = {(f(x), g(x)) | x ∈ ℝⁿ}, f(x) = x^TAx + 2a^Tx + a₀ and g(x) = x^TBx + 2b^Tx + b₀. We show that R(f, g) is convex if, and only if, any pair of level sets, {x ∈ ℝⁿ|f (x) = α} and {x ∈ ℝⁿ|g(x) = β}, do not separate each other. With the novel geometric concept about separation, we provide a polynomial-time procedure to practically check whether a given R(f, g) is convex or not.