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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?

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up vote 9 down vote accepted

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 $z'' = (x'',y'')$.

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Thank you so much. – Supasate Aug 5 '11 at 11:47

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