# Convex Functions

*Definition. Convex function:* A function $f:X_D \rightarrow X_R$ with
$X_D\subseteq\RR^n$ and $X_R\subseteq\RR$ is a convex function if for any $x_1$
and $x_2$ in $X_D$ and $\lambda_1 \geq 0$, $\lambda_2 \geq 0$ such that
$\lambda_1+\lambda_2=1$, we have:
\begin{equation}
f(\lambda_1x_1+\lambda_2x_2)\leq \lambda_1f(x_1)+\lambda_2f(x_2)
\end{equation}