Convexity of a small ball under quadratic map

We derive an upper bound on the size of a ball such that the image of the ball under quadratic map is strongly convex and smooth. Our result is the best possible improvement of the analogous result by Polyak [1] in the case of a quadratic map. We also generalize the notion of the joint numerical range of m-tuple of matrices by adding vector-dependent inhomogeneous term and provide a sufficient condition for its convexity.