Convexity of "perturbed" convex function

Question

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!

Answer 1

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

Answer 2

This fails already in dimension 1. For any $A,B,\epsilon$, you can find $C\gg 1$ such that

$$ f(x) = Ax^2 + Bx + x \epsilon \sin(Cx)$$

is not convex.