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  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.
- Joint numerical range
- Quadratic transformation (map)
- Trust region problem