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Standard simplex

Definitions

The standard n-simplex is the set

$\Delta^n=\{(x_1,...,x_{n+1}) \in \mathbb{R}^{n+1}: x_i \ge 0 \forall i \land \sum_i x_i = 1\}$

For each nonempty subset $A \subseteq \{1,...,n+1\}$ there is a corresponding face:

$\{(x_1,...,x_{n+1}) \in \Delta^{n}: x_i = 0 \forall i \notin A\}$

Properties

Note that the simplex is closed.

n is the dimension of the simplex

Its vertices are $V(\Delta^n) = \{(x_1, ..., x_{n+1}): x_i \in \{0,1\}: \sum x_i = 1\}$.

The inside of $\Delta^n$ is

inside$(\Delta^n) = \{(x_1,...,x_{n+1}) \in \Delta^{n}: x_i > 0 \forall i\}$

Note that the inside of $\Delta^0$ is $\Delta^0$.