Convex inequality on Hilbert space

Question

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.

This inequality is true for all $v \in H$ by convexity. You do not need that $v$ is a minimizer. — gerw, Sep 19 '16 at 20:23
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 $v$, in holds for every point. — dohmatob, Sep 20 '16 at 08:15
@dohmatob Could you help me with the proof of that inequality, please? — Tien Kha Pham, Sep 20 '16 at 11:33

score 1 · Accepted Answer · answered Sep 20 '16 at 12:03

1

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.

answered Sep 20 '16 at 12:03

dohmatob

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Convex inequality on Hilbert space

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