If the sum of two functions is convex is at least one of them also convex?

Question

I know that the sum of two convex functions $f$ and $g$ is convex, but can we also state the opposite? If the sum $f+g$ is convex, does it mean that at least one of them is convex?

No. $\exp x + \sin x$ is not convex, nor is $-\sin x$, but their sum is. And so on. — kimchi lover, Feb 03 '19 at 19:13

score 2 · Accepted Answer · answered Feb 03 '19 at 19:19

2

No, not necessarily.

Let $f(x) = \sqrt{x}$ on $[0,1]$ and $x-\sqrt{x}+1$ on $[1,2]$

Let $g(x) = x-\sqrt{x}$ on $[0,1]$ and $\sqrt{x}-1$ on $[1,2]$.

Then $f+g = x$ which is convex but neither $f$ nor $g$ are convex.

answered Feb 03 '19 at 19:19

Mike

20,434

2

If you want strictly convex replace $x$ with (say) $x^2$ – Mike Feb 03 '19 at 19:22

If the sum of two functions is convex is at least one of them also convex?

1 Answers1