Let $f=f(x,y)\in C^1(\Omega)$ for some convex domain $\Omega\in\mathbb{R}^2$. Suppose $x\mapsto f(x,y)$ is convex $\forall y$ possible and $y\mapsto f(x,y)$ is also convex $\forall x$ possible. Is $f$ convex on $\Omega$?
I think it is not true, but cannot come up with proper counterexample.
Example: $f(x,y) = xy$.
For each fixed $y$ it is linear in $x$ and therefore convex in $x$. For each fixed $x$, similarly.
But along the line $x+y=0$ it is not convex.
$f(1,-1) = -1$, $f(-1,1) = -1$, but the midpoint $f(0,0) = 0$ is not below $-1$.