What is Convex set, convex function, convex optimisation problem?

A set $C \subseteq \R^n$ is a convex set if it contains the segment between any two of its points. A function $f$ is a convex function if $f(\theta x + (1 - \theta)y) \le \theta f(x) + (1 - \theta)f(y)$ for $\theta \in [0, 1]$. A convex optimisation problem minimises a convex function subject to convex inequality constraints $g_i(x) \le 0$ and affine equality constraints $Ax = b$.