Let $H$ be a Hilbert space, $\phi: H \to \mathbb{R}$ be a convex function which is bounded below, $\phi \in C^1$ and $\nabla \phi$ is locally Lipschitz. Suppose there exists $v$ in $H$ such that $\phi(v) = \min \phi$. Then we have the convex inequality $$\phi(v)\ge \phi(x) + \langle \nabla \phi (x), v- x\rangle, \quad \forall x\in H.$$
I want to ask whether this inequality is true or false. If is is true, how can we prove it? If it is false, which additional conditions are neccessary?
Thank you so much.
I agree with gerw. Your inequality is nothing but the characterization of convexity for $\mathcal C^1$ functions. You don't need any conditions on $v$; it holds for every point.
For a proof, see section 2 of this handout.