Suppose that $f$ is a weighted average of the functions $g_i$, $$f(x) = \sum_{i=1}^N \lambda_i \cdot g_i(x)$$ (where each $\lambda_i >0$). Under what conditions is the maximum value of $f$ a weighted average of the maximum values of the $g_i$? $$ \max_x f(x) = \sum_{i=1}^N \lambda_i \cdot \max_x g_i(x)$$
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Are the $\lambda_i$ non-negative? – David G. Stork Sep 03 '18 at 17:47
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Yes. I've updated the question. – JDG22 Sep 03 '18 at 17:49
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@joriki why is it not sufficient? Is that easy to see? – Theoretical Economist Sep 03 '18 at 18:07
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@joriki yes, I understand that that would have to be the explanation, but I just don’t see it. I’ll just give it more thought. Thanks. – Theoretical Economist Sep 03 '18 at 18:38
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1@TheoreticalEconomist: Sorry, it was I who was confused :-) I deleted the comment. – joriki Sep 03 '18 at 18:40
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@JDG22: Are you looking for sufficient or necessary conditions, or both? – joriki Sep 03 '18 at 18:42
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@joriki yes, I was thinking convexity of the max operator would be enough to show that the condition you stated was sufficient. I doubt it is necessary, however. – Theoretical Economist Sep 03 '18 at 18:42
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Unless all $g_i(x)$ have the maximum at the same $x$ it is hard to see if it is true. – herb steinberg Sep 03 '18 at 19:51