Convexity of "perturbed" convex function Suppose $x\in\mathbb{R}^n$ and that we have the following "perturbed" quadratic form: $$Q(x) = x^\intercal A x + x^\intercal B + x^\intercal F(x)$$ where $A\in\mathbb{R}^{n\times n}$, $(A+A^\intercal)$ is positive-definite, $B\in\mathbb{R}^n$ and the nonlinear mapping $F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is bounded, i.e. $\Vert F(x) \Vert \leq \gamma$. I'm interested in the following question: _Old question: Is there any $\gamma\in\mathbb{R}$ such that $Q(x)$ retains convexity under the presence of $F(x)$?_ **_New edited question: Can I find a condition on $\gamma\in\mathbb{R}$ such that I can guarantee convexity of $Q(x)$ under the presence of $F(x)$?_** Thanks in advance!

No, simply let $F(x) = -2Ax$ when $||2Ax||\leq \gamma$ (and 0 otherwise) and you will always have a negative definite quadratic locally somewhere.