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Suppose $f$ is the sum of a concave and convex function, i.e. $$f=f_1+f_2$$ where $f_1$ is a concave function and $f_2$ is a convex function. I wonder if $f$ can be written as the following: $$f=\min\{g_1,g_2\}$$ for some concave function $g_1$ and convex function $g_2$.

I try to write the minimum as $$f_1+f_2=f=\min\{g_1,g_2\}=\frac{g_1+g_2-|g_1-g_2|}{2}$$ and express $g_1$ and $g_2$ in terms of $f_1$ and $f_2$. But it seems to me that it does not help much.

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    $\begingroup$ If you can do this to $f$, you can do this to any $C^2$ function (on an interval), as proved here. $\endgroup$ – vadim123 Jul 1 '13 at 1:46
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    $\begingroup$ If $f=\min(g_1,g_2)$, then $g_1(x)\ge f(x)$ for all $x$. I'm pretty sure there is no concave $g_1(x)$ which lies above $f(x)=x^2$ everywhere. $\endgroup$ – Rahul Jul 1 '13 at 2:06

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