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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?

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As each summand is concave, so f is concave. A relevant problem is when a product of quadratic form is convex. A reference comes to me is the paper "Lin, Sinnamon, A condition for convexity of a product of positive definite quadratic forms, SIAM J. Matrix Anal. Appl. 32 (2011) 457-462." –  Betrand May 7 '13 at 12:00
It is neither convex nor concave (ignore for now the extra constraint that $x^Tx=1$, because with that constraint, you are asking for convexity on the surface of a hypersphere, which can at best hold only very locally); to see why, simply generate a few random vectors and test what happens to $f$. –  Suvrit May 8 '13 at 22:02
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