If $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is a differentiable convex function, then we know that its gradient is monotone: $$ r(x,y) := (x-y)^T(\nabla f(x)-\nabla f(y)) \geq 0\,, \quad \text{for all} \quad x,y \in \mathbb{R}^n $$ I am interested in the function $r(x,y)$. By inspection, the following three properties hold:
- $r(x,y) \geq 0$
- $r(x,x) = 0$
- $r(x,y) = r(y,x)$.
General question: what else interesting can be said about the function $r(x,y)$?
Specific question: Does (or when does) $r(x,y)$ admit a factorization $$ r(x,y) = (h(x)-h(y))^{\sf T}(h(x)-h(y)) $$ for some vector-valued mapping $h$?