Theorem 31.6:

If ${\bf v}_1, {\bf v}_2, \dots, {\bf v}_k \in \mathbb{R}^n$ are affinely independent then every point $x$ in the convex hull of ${\bf v}_1, {\bf v}_2, \dots, {\bf v}_k$ can be uniquely expressed as

$$x = t_1{\bf v}_1 + t_2{\bf v}_2 + \dots + t_k{\bf v}_k, \eqno(31.2)$$

where the coefficients $t_j$ ( $1\leq j\leq k$) are non-negative and $t_1 + t_2 + \dots + t_k = 1$. These coefficients are called the barycentric coordinates of $x$.

We consider the standard $n$ simplex $\Delta_{n}$ in $\mathbb{R}^{n+1}$ with summit $S = {\bf e}_{n+1}$. The figure below depicts a general point $Q$ on the face $\Delta_{n-1}$ opposite to $S$ and $P$ an arbitrary point on the line segment joining $Q$ and $S$. The reader may check that if $\lambda_1, \lambda_2, \dots ,\lambda_{n+1}$ are the barycentric coordinates of $P$ then the coordinates of $Q$ are given by the $n-$tuple

$$U(\lambda_1, \dots, \lambda_{n+1}) = \Big( \frac{\lambda_1}{1-\la... ...}{1-\lambda_{n+1}}, \dots, \frac{\lambda_n}{1-\lambda_{n+1}} \Big) \eqno(31.3)$$

Note that $U$ is bounded but not continuous when $\lambda_{n+1}\longrightarrow 1$. in As $P$ approaches $S$ the pyramid with base $\Delta_{n-1}$ and summit $P$ fills up $\Delta_n$. We are now in a position to prove the following theorem.