Simplex

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(pl. simplices). LaTeX: \textstyle \left \{ x \in \mathbb{R}: \sum x_j = 1 \right \}. For LaTeX: n=1, this is a point LaTeX: (x=1). For n=2, this is a line segment, joining points (1,0) and (0,1). For LaTeX: n=3, this is a triangle, joining the vertices (1,0,0), (0,1,0), and (0,0,1). This is sometimes called an n-simplex, denoted by LaTeX: S_n (note its dimension is n-1). The open simplex excludes the axes: LaTeX: \textstyle \{ x \in S_n: x > 0\}.

More generally, some authors define a simplex to be the convex hull of any LaTeX: n+1 affinely independent vectors, and refer to the special case of the unit vectors as the standard simplex.


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