Today I Learned

Some of the things I've learned every day since Oct 10, 2016

90: The Standard (Unit) Simplex

The standard n-simplex, sometimes denoted \Delta^n, is the n-simplex in \mathbb{R}^{n + 1} with its vertices at the n + 1 ‘unit’ points

e_0 = (1, 0, 0, \dots, 0)

e_1 = (0, 1, 0, \dots, 0)

e_2 = (0, 0, 1, \dots, 0)

\dots

e_n = (0, 0, 0, \dots, 1).

For example, \Delta ^0 is the point 1 in \mathbb{R}^1\Delta ^1 is the line segment from (0, 1) to (1, 0) in \mathbb{R}^2, and so on. A formal description is

\Delta ^n = \{(x_0, x_1, \dots, x_n) \in \mathbb{R}^{n+1} | \sum _{i = 0} ^{n} x_i = 1, x_i \geq 0 \}.

One possible interpretation of \Delta ^n is as the set of possible probability distributions of a categorical distribution on n+1 variables.

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