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Suppose $f:\mathbb{R}^n\mapsto \mathbb{R}$ is convex, could anyone tell me how to prove the following fact?

(1) If $f\in C^1$, then for any $u,v$

$$ f(v)\geqslant \langle\nabla f(u),v-u\rangle $$

(2) If $f\in C^2$, then for any $u,v$ $$ \langle\nabla^2f(u)v,v\rangle\geqslant0 $$

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Useful fact: the restriction of a convex function to a line is a convex function.

Formally, for any point $a$ and any nonzero vector $w$ the function $\varphi(t) = f(a+tw)$ is convex on $\mathbb R$.

  • If $f$ is $C^1$, then so is $\varphi$. We have $\varphi(t)-\varphi(0)=\varphi'(\xi)\,t>\varphi'(0)\,t$ by the Mean Value Theorem and because $\varphi'$ is increasing. In terms of $f$ this translates into the inequality you want.
  • If $f$ is $C^2$, then so is $\varphi$. Consequently, $\varphi''(0)\ge 0$ which translates into the inequality you want.
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