Let $A_i$, $i=1,\dots,L$ be given $N\times N$ positive definite real matrices. I have this sum of exponentials \begin{align} f(\mathbf{x})=\sum_{i=1}^{L}\operatorname{exp}(-{\mathbf{x}^T\mathbf{A}_i\mathbf{x}}),~~~~~~\mathbf{x}\in \mathbb{R}^N,\mathbf{x}^T\mathbf{x}=1 \end{align}

Has this function been studied before. Can someone point me to relevant references?. Or anyone can make some comment on it as if it is convex or concave?