Sum of Quasiconvex Functions on Different Domains

Question

Say $f(x)$ and $g(y)$ are both quasiconvex functions. Is the function $h(x,y) = f(x) + g(y)$ also quasiconvex? I know sums of quasiconvex functions on the same domain are not necessarily quasiconvex, but what is the case on different domains?

Answer 1

No, consider $f(x) = \sqrt(x)$ and $g(y) = \sqrt{y}$. The set $\{(x,y) : \sqrt{x} + \sqrt{y} \leq 1\}$ is not convex: $(1,0)$ and $(0,1)$ are in the set, but $(0.5, 0.5)$ is not.