Convex function

a convex set is essentially a set that intersects every line in a line segment ¹

ELI5: a convex function graph is shaped like a cup $\cup$, where as a concave function graph is shaped like a cap $\cap$

formal definition

Let $X$ be a convex subset of a real vector space and let $f: X \to \mathbb{R}$ be a function.

Then $f$ is called convex iff any of the following equivalent holds:

1. $\forall 0 \le t \le 1 \cap x_{1}, x_{2} \in X \mid f(tx_{1}+(1-t)x_{2}) \le tf(x_{1}) + (1-t)f(x_{2})$
2. $\forall 0 \le t \le 1 \cap x_{1}, x_{2} \in X \text{ where } x_{1} \neq x_{2} \mid f(tx_{1}+(1-t)x_{2}) \le tf(x_{1}) + (1-t)f(x_{2})$

This is also known as the Jensen’s inequality:

probability theory

The form as follows:

Given $X$ is a random variable with $\varphi$ a convex function, then

$$\varphi(E[X]) \le E[\varphi{(X)}]$$