Definition:Simplex/Definition 2

Definition
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:
 * $\forall i \in \set {0, 1, 2, \ldots, n}: \theta_i \in \closedint 0 1$
 * $\ds \sum_{i \mathop = 0}^n \theta_i = 1$

Also see

 * Equivalence of Definitions of Simplex


 * Definition:Pentatope