\begin{definition}[Definition:Simplex/Definition 2]
A '''simplex''' $S$ in $\R^n$ with vertices $\family {\alpha_i}_{i \mathop = 0}^n$ is a set such that:
:$S = \set {\ds \sum_{i \mathop = 0}^n \theta_i \alpha_i}$
where:
:$\sequence {\alpha_i}_{i \mathop = 0}^n$ is a sequence of $n + 1$ affinely independent points in $\R^n$
:$\sequence {\theta_i}_{i \mathop = 0}^n$ is a sequence of arbitrary real numbers such that:
::$\forall i \in \set {0, 1, 2, \ldots, n}: \theta_i \in \closedint 0 1$
::$\ds \sum_{i \mathop = 0}^n \theta_i = 1$
\end{definition}