On the Convexity of Image of a Multidimensional Quadratic Map

We study convexity of image of a general multidimensional quadratic map. We split the full image into two parts by an appropriate hyperplane such that one part is compact, and formulate a sufficient condition for convexity of the compact part. We propose a way to identify such convex parts of the full image which can be used in practical applications. By shifting the hyperplane to infinity we extend the sufficient condition for convexity to apply to the full image of the quadratic map. As a related result, we formulate a novel condition for the joint numerical range of m-tuple of hermitian matrices to be convex. Finally, we illustrate our findings by considering several examples. In particular we prove convexity of solvability set for the Power Flow equations in case of DC networks.