Is the sum of convex functions on different domains convex?
Solution 1:
If you write $$ h(az'+(1-a)z'') = h(ax'+(1-a)x'',ay'+(1-a)y'') $$ $$= f(ax'+(1-a)x'')+g(ay'+(1-a)y'') $$ $$\leq af(x') +(1-a)f(x'')+ag(x')+(1-a)g(x'') $$ $$= ah(z')+(1-a)h(z'') $$ you will see that $h$ is convex. Here $z' = (x',y')$ and $z'' = (x'',y'')$.