Convexity of a Small Ball Under Quadratic Map

TL;DR

This paper establishes optimal bounds for the convexity of images of small balls under quadratic maps and generalizes joint numerical range concepts, providing conditions for convexity in matrix analysis.

Contribution

It improves existing bounds on convexity of quadratic map images and introduces a generalized joint numerical range with convexity conditions.

Findings

Derived the best possible upper bound for convexity of quadratic map images.

Generalized joint numerical range with inhomogeneous terms.

Provided sufficient conditions for convexity of the generalized range.

Abstract

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 quadratic map. We also generalize the notion of the joint numerical range of m-tuple of matrices by adding vector-dependent inhomogeneous terms and provide a sufficient condition for its convexity.