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Let $C$ be the smallest closed convex cone of functions from $\mathbb{R}^n$ to $\mathbb{R}$ that contains all constant functions, all coordinate functions, and such that $\max(f,0)\in C$ whenever $f\in C$.

Is there any useful characterization of the functions lying in $C$? It is easy to see that such functions must be convex and in addition have non-negative partial derivatives up to the second order, but I would think these conditions aren't sufficient.

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    $\begingroup$ The cone only contains coordinate functions, not their opposite (if you include opposite coordinate functions the cone includes all convex functions) $\endgroup$
    – alesia
    Commented Sep 11, 2022 at 17:22
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    $\begingroup$ Also if you add two different functions of the form you described, you might get a function that is not of this form $\endgroup$
    – alesia
    Commented Sep 11, 2022 at 17:24
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    $\begingroup$ You're assuming that the cone is closed with respect to which topology? $\endgroup$
    – YCor
    Commented Sep 11, 2022 at 18:04
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    $\begingroup$ @YCor let's say pointwise convergence $\endgroup$
    – alesia
    Commented Sep 11, 2022 at 18:14
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    $\begingroup$ @Ycor yes that's correct. Starting from two dimensions things seem to get more complicated $\endgroup$
    – alesia
    Commented Sep 11, 2022 at 19:26

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