Convex sets and barycentric coordinates:

Let ${\bf v}_1, {\bf v}_2, \dots, {\bf v}_k \in \mathbb{R}^n$ be given points. The convex hull of these points is the set consisting of all convex combinations $t_1{\bf v}_1 + t_2{\bf v}_2 + \dots + t_k{\bf v}_k$, that is the coefficients $t_1, t_2, \dots, t_k$ are non-negative and $t_1 + t_2 + \dots + t_k = 1$. The convex hull of these points is clearly a convex set. The points ${\bf v}_1, {\bf v}_2, \dots, {\bf v}_k \in \mathbb{R}^n$ are said to be affinely independent if the $k-1$ vectors ${\bf v}_1 - {\bf v}_k, {\bf v}_2 - {\bf v}_k, \dots, {\bf v}_{k-1} - {\bf v}_k$ are linearly independent (see exercise 4). The convex hull of a set of $k$ affinely independent points is called the affine $k-$ simplex spanned by these points. The proof of the following result is left as an exercise.