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Here's a homework question I'm struggling with:

Let $f,g$ two convex functions. Prove that $h(x)=\max\{f(x),g(x)\}$ is also convex

I don't know where to begin. The only thing I had in mind was was to try proving that if a function is convex on two sets $A$ and $B$, it is also convex on their union. That does not seem right though, for example if I glue together $f(x)=x^2, g(x)=\frac{x^2}{1000}$ where $f$ is defined on $[0,1]$ and $g$ on $(1,2]$.

Anyway, that was the only thing I thought about. Any better ideas? thanks!

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@yotamoo: While typesetting, use \max instead of max (similarly use \sin instead of sin etc) – user17762 May 20 '12 at 17:54
@Didier Can you add that as an answer adding, probably another line or two, so that this question gets an answer? – user17762 May 20 '12 at 18:04
up vote 3 down vote accepted

Hint: Use the characterization that $h$ is convex if and only if, for every $t$ in $[0,1]$ and every $(x,y)$, $h(tx+(1-t)y)\leqslant th(x)+(1-t)h(y)$.

Second hint: One wants to prove that $h(z)\leqslant th(x)+(1-t)h(y)$ where $z=tx+(1-t)y$. Since $h=\max\{f,g\}$, this is equivalent to the two inequalities $$ f(z)\leqslant th(x)+(1-t)h(y),\qquad g(z)\leqslant th(x)+(1-t)h(y). $$ Consider the first inequality. By convexity of $f$, one knows that $f(z)\leqslant tf(x)+(1-t)f(y)$. Furthermore, $f(x)\leqslant$ $____$ and $f(y)\leqslant$ $____$, hence...

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Okay I can assume that $h=g$ at $tx_1+(1-t)x_2$, but the behavior of $h$ at $x_1$ and $x_2$ has four different options, depending on what function is greater at these points. How can I come to a conclusion? – yotamoo May 21 '12 at 7:08
Nevermind I saw it, thanks! – yotamoo May 21 '12 at 7:31

The hint of @Did solves the problem, but there is another proof, which is more intuitive I think.

A function is convex if and only if the area above its graph is convex. But then, the region above $h(x) = \max\{f(x),g(x)\}$ is the intersection of the area above $f$ and the region above $g$. Moreover, intersection of convex sets is convex, and that concludes the proof.

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Does this apply similarly for the maximum of two concave functions? – Trevor Alexander Apr 18 '14 at 5:02
@TrevorAlexander Not really, take for example $f(x) = -(x-2)(x-4)$ and $g(x) = -(x+2)(x+4)$, then $h(-4) = h(-2) = h(2) = h(4) = 0$, but $h(-3) = h(3) = 1$, hence $h$ is not concave. – dtldarek Apr 18 '14 at 7:27

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