Q: Is it a convex function?

Question

Let $f(v):= g(vv^t)$ with $v \in \mathbb{R}^2$.

If then the function $g: \mathbb{R}^{2\times 2} \rightarrow \mathbb{R}$ is defined as $g(X) = X_{12} + X_{21}$, will the function $f$ be convex over the vectors $v$?

Answer 1

No. $f(v) = 2 v_1 v_2$, which is well known to be neither convex or concave, as its Hessian has a positive eigenvalue and a negative one.