I'm interested in the Kachurovskii's theorem in the Euclidean case: Let $m \ge 1$ be an integer and let $g: \mathbb{R}^m \to \mathbb{R}^m$ be a function that is an increasing monotone operator, i.e.,
$$(g(x) - g(y))^\top (x-y) \ge 0$$ for all $x, y \in \mathbb{R}^m$.
Question. Is $g$ necessarily the derivative of some convex $f : \mathbb{R}^m \to \mathbb{R}$? In other words, does Kachurovskii's theorem characterizes all increasing monotone operators?
