Let $g: \mathbb{R}^n\times\mathbb{R}^m \rightarrow \mathbb{R}$. Is there a condition under which $$\inf_{y\in\mathbb{R}^m} [g(x_1,y) + g(x_2,y)] = \inf_{y\in\mathbb{R}^m} [g(x_1,y)] + \inf_{y\in\mathbb{R}^m}[g(x_2,y)],$$
where $x_1,x_2 \in \mathbb{R}^n$?
I was wondering whether convexity of $g$ is enough for the condition to hold but I'm not able to prove it.
