# Is the sum of multiple convex functions convex?

So in my textbook it says that if $$f$$ and $$g$$ are convex functions, then $$h = f + g$$ is also a convex function.

My question is, does this extend to a sum of more than just two convex functions. If $$f_1,f_2,\dots,f_n$$ are all convex functions, is the function $$h = f_1 + f_2 + \dots + f_n$$ also a convex function?

My attempt at a non formal proof, was that if you let $$f$$ and $$g$$ be defined as the sum of two convex functions, than surely you could extend the sum by simply letting your functions be defined as the sum of two convex functions.

• Isn't this just a typical proof by induction? Where are you stuck? Apr 10, 2021 at 14:32
• Induct on the number of summands. Apr 10, 2021 at 14:32

## 1 Answer

Proof that sum of two convex function is a convex function is here Prove that the sum of convex functions is again convex.

On the other hand, if you have $$n$$ functions, you know that sum of first two is a convex. Then, You repeat this argument by adding $$(f_1 + f_2)+f_3$$ where the first bracket is a convex function. Therefore yes, this holds for any finite sum. It is an induction principle too.

• Thanks, yeah that makes perfect sense, just wanted to double check. Apr 10, 2021 at 14:43