It's well-known that negative-log-determinant is convex, i.e. $- \log \det X = \log \det X^{-1}$ ($x \in \mathbf{S}^n_{++}$) is convex thus $\det X^{-1}$ is log-convex so it must be convex.
This is a simple result, but I have never seen it anywhere. Instead, I always see negative-log-determinant. And I don't know why. Is it because the gradient of $\det X^{-1}$ still involve $\det X^{-1}$ but add a log operation outside can cancel this determinant term?
