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Under which conditions is the union of conic hulls of sets in a cartesian product equal to $\mathbb{R}^N$?

Question: Under which conditions on $A, B\in\mathbb{R}^{N\times N}$ is the function $f: \mathbb{R}^N\mapsto \mathbb{R}^N$, $$f(v) = A[v]_+ + B[-v]_+$$ surjective? Here $[.]_+$ is an elementwise ...
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