Simplices

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points ${\displaystyle u_{0},\dots ,u_{k}\in \mathbb {R} ^{k}}$ are affinely independent, which means ${\displaystyle u_{1}-u_{0},\dots ,u_{k}-u_{0}}$ are linearly independent. Then, the simplex determined by them is the set of points

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