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Define function $f^i := h^i - g^i$ where $f^i$, $g^i$ are real valued convex functions for all $i = 1, ..., 10$.

Does there exist real valued convex functions $h$, $g$ where $h - g = \max_{i} f^i$?

I don't even know how to even approach this problem.

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Yes, the class of DC functions (difference-convex, or delta-convex as they are sometimes called) is a lattice: it's closed under taking pointwise max or min. It suffices to show that for any two DC functions $f_1, f_2$ their maximum $\max(f_1, f_2)$ is also DC.

Note that $\max(f_1, f_2) = \frac12|f_1+f_2| + \frac12|f_1-f_2|$, where both $f_1 \pm f_2 $ are DC. Thus, it suffices to show that the absolute value of a DC function is DC.

If $f$ is DC, then from $f = h-g$ we get $\max(f, 0) = \max(h, g) - g$, hence $\max(f, 0)$ is also DC.

And then $|f| = \max(f, 0) + \max(-f, 0)$ is DC as well.

This proof, and much more, can be found in On difference convexity of locally Lipschitz functions by Miroslav Bačak and Jonathan M. Borwein.

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  • $\begingroup$ Is it possible, in general, to determine $h$ and $g$ based on all the $h^i$ and $g^i$? $\endgroup$ – master_goon Jul 23 '18 at 21:58
  • $\begingroup$ Yes, by following the steps given above. It's all constructive. Of course, $h, g$ are not uniquely determined: one can add any convex function to both of them. $\endgroup$ – user357151 Jul 23 '18 at 22:19

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