Definition (Convex set)

A subset $K$ of $\mathbb{R}^n \ (K\subset\mathbb{R}^n)$ is called convex if the line segment joining any two points in $K$ also lies in $K$. Given two points $\vec x_1,\vec x_2\in K$, the line segment joining $\vec x_1$ and $\vec x_2$ is defined as $$L_{\vec x_1\vec x_2}=\{\vec x\in\mathbb{R}^n: \ \vec x=\lambda\vec x_1+(1-\lambda)\vec x_2, \ \lambda\in[0,1]\}$$ which can also be understood as the set of **all convex combinations of $\vec x_1$ and $\vec x_2$.