# What is the closed cone generated by constant and coordinate functions and closed under taking $f\mapsto\max(f,0)$?

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.

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