Consider the functions $f:\mathbb{R}^n\to\mathbb{R}$, $g:\mathbb{R}^m\to\mathbb{R}$ and the non-square matrix $A \in \mathbb{R}^{m\times n}$ with $m>n$.
Let $x\in\mathbb{R}^n$ and consider the change of variables $y = Ax$. Let $f(x) = g(Ax)$.
If $g(y)$ is strictly convex in $y$, under what conditions on $A$ can we say that $f(x)$ is strictly convex in $x$?
My attempt:
The Hessian of $g$ is positive definite, that is $H_g\succ0$. Is the Hessian of $f$ equal to $A^TH_gA$?
Would $A$ being full rank ensure that $H_g\succ0\Rightarrow A^TH_gA\succ0$?