I need help proving that if $g(x)=e^{\langle a,x\rangle }*f(x)$ is convex on $\mathbb{R}^n$ for all $a \in \mathbb{R}^n$, then $f$ is logarithmically convex.
One thing I noticed was that if $g$ is log-convex itself, then $f$ is log-convex since then, $\log(g) = \langle a,x\rangle + \log(f)$, so that $\log(f) = \log(g)-\langle a,x\rangle$. Then $\log(g)$ would be convex, and $-\langle a,x\rangle$ is a linear function so is also convex, and hence their sum is as well.
However, I can't seem to prove this either. Other things I thought may be helpful are that $g$ is convex iff its hessian is positive semidefinite, or iff its gradient is monotone, but neither of these have been particularly useful to me.
