I have a constraint looks like $$g^T(FP^{-1}F^T)^{-1}g>1$$ where $P\in S_{++}^{n\times n}$ $g\in\Re^{m}$ ,$F\in \Re^{m\times n}$ ,$n>m$ and $P$ is a variable.
When $m=1$, it can be convert to convex constraint by Schur complement easily.
However, when $m>1$, I can only get result like $$ det\begin{bmatrix} P & F^T \\ F & g\times g^T \end{bmatrix}>0 $$ It is neither a convex nor concave.
At first step of Schur complement, as it's $$1 - g^T(FP^{-1}F^T)^{-1}g<0$$, the results about positive definiteness can not be applied.
Is there any other methods can be applied to this question ? Is it possible to convert this constraint to a semidefinite constraint?