Let
$$ \Delta_n := \left\{x\in \mathbb{R}^n:\sum_{i=1}^n x_i = 1, x_i \in [0,1] \right\}$$
Let $f, G$ be linear functions. For any $x\in \Delta_n$, $G$ is a positive definite matrix. I am trying to prove the strong convexity of the function $F : \Delta_n \to \mathbb{R}$ defined as follows
$$ F(x) := f(x)^{\top}G(x)^{-1}f(x),$$
The Hessian is too complex to be calculated, so I tried to prove by the definition of strong convexity. i.e., $$ F(\alpha x_1+(1-\alpha)x_2)\leq \alpha F(x_1)+(1-\alpha)F(x_2)-c\|x_1-x_2\|^2,~~ \forall x_1,x_2,\text{ and }\forall\alpha\in [0,1]. $$
However I got stuck when expanding the terms, particularly with the term $$(\alpha G(x_1)+(1-\alpha)G(x_2))^{-1}.$$
How can I proceed? Or will $F$ not be strongly convex?
