Let $E \subseteq \mathbb{R}^k$ be a convex set. A function $\varphi : E \to \mathbb{R}$ is convex if

$$ \varphi\big(\lambda \vec{x} + (1 - \lambda) \vec{y}\big) \leq \lambda \varphi(\vec{x}) + (1 - \lambda)\varphi(\vec{y}) $$

whenever $\vec{x}, \vec{y} \in E$ and $0 < \lambda < 1.$

In geometric terms, this means a @chord joining any two points on the graph of $\varphi$ lies on or above the graph between them.

Starting with convex function, $\varphi$ is strictly convex if the inequality is strict whenever $\vec{x} \neq \vec{y}.$

A function $\varphi$ is said to be concave if $-\varphi$ is a convex function.