Let $\mathbf{A}$ and $\mathbf{b}$ be a symmetric $N\times N$ real matrix and $N\times 1$ real vector respectively. Then consider the set of points in $\mathbb{R}^2$ defined as \begin{align} \mathbb{S}=\{\left(\mathbf{u}^T\mathbf{A}\mathbf{u},\mathbf{b}^T\mathbf{u}\right)\in\mathbb{R}^2,\mathbf{u}^T\mathbf{u}=1\} \end{align} That $\mathbb{S}$ is bounded is obvious. I was wondering if it had any other properties. In particular I am looking at convexity, similar to numerical ranges. If not, what are the conditions on $\mathbf{A}$ and $\mathbf{b}$ for the convexity and so on.

Motivation: Consider the optimization problem \begin{align} &\min_{\mathbf{u}\in \mathbb{R}^N}\mathbf{u}^T\mathbf{A}\mathbf{u}+\mathbf{b}^T\mathbf{u} \\\ \mbox{subject to }\\\ &\mathbf{u}^T\mathbf{u}=1 \end{align} Then using the definition of $\mathbb{S}$ as earlier, I can write the above problem as \begin{align} \min_{(x,y)\in\mathbb{S}}x+y \end{align} Thus if $\mathbb{S}$ is convex, the above problem will be convex.