I just started learning about nonlinear optimization, and I'm having trouble understanding concepts of convex. Could anyone help me write and understand the proof of the below statement? Thank you!
Let $g :P → \mathbb{R}$ be a continuous function on a convex domain $P⊆\mathbb{R}^n$. Show that g is convex if and only if $$ (a-b)^T (∇g(a)−∇g(b)) \ge 0, \quad ∀ a,b ∈ P $$