I have a fixed invertible matrix, ${\bf X} \in \mathbb{R}^{n\times n}$, and a vector $c\in[0,1]^n$ (i.e each element of $c$ is between $0$ and $1$). I have numerical evidence that the following function is a convex function of $c$ in this specific range, no matter the value of ${\bf X}$:
\begin{equation} f(c) = \sum_{ij=1}^n\frac{X_{ij}^2 + (X_{ij}^{-1})^2 - 2c_ic_jX_{ij}X_{ij}^{-1}}{1-c^2_i} \end{equation}
where $X_{ij}^{-1}$ is the $ij$-th element of the inverse of ${\bf X}$.
Unfortunately I cannot prove it is convex, and I was looking for advice; how would you go about this? Can you prove it is convex?
I've tried looking at the Hessian and showing it is positive semi-definite, or just trying to show that $f(\lambda c + (1-\lambda)c') \leq \lambda f(c) + (1-\lambda)f(c')$, but no dice. Are there other classic techniques? Or am I being dumb and missing an obvious reason it either is or isn't convex?