Convex functions in probability: $h(\alpha x_1 + (1 - \alpha)x_2) \le \alpha h(x_1) + (1 - \alpha)h(x_2)$
$f \text{ convex }\Rightarrow \forall (x_0,f(x_0))\in \Gamma(f) \exists a : \forall x\in \mathbb{R}, f(x) \ge a(x-x_0)+f(x_0)$