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.

  • 3
    $\begingroup$ Isn't this just a typical proof by induction? Where are you stuck? $\endgroup$
    – hardmath
    Apr 10, 2021 at 14:32
  • $\begingroup$ Induct on the number of summands. $\endgroup$ Apr 10, 2021 at 14:32

1 Answer 1


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.

  • $\begingroup$ Thanks, yeah that makes perfect sense, just wanted to double check. $\endgroup$
    – jakvah
    Apr 10, 2021 at 14:43

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