On the same domain, the sum of convex functions is convex (e.g. $f(x) + g(x)$ is convex if $f(x)$ and $g(x)$ are convex). However, I don't know that this is true for the sum of convex functions on different domains.
For example, let $f(x) | x \in \mathbb{R}^n $ is convex and $g(y) | y \in \mathbb{R}^n $ is also convex, is $h(x,y) = f(x) + g(y)$ convex?
If it is not convex, I would like to know further that is the sum of the "same convex functions" on "different domains" convex? For example, if $f(x)$ is convex, is $h(x,y) = f(x) + f(y)$ convex?
