Convex Function

Let $X$ be a convex subset of a real vector space and let $f: X \rightarrow \mathcal{R}$ be a function. Then $f$ is called convex if and only if

For all $0 \leq t \leq 1$ and all $x_1, x_2 \in X$:

\[ f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) \]