  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.
