### The Simplex

The simplex is a generalization to n-dimensions of the triangle in 2D and the tetrahedron in 3D. An n-simplex is the convex hull of n+1 points that do not all lie in some n−1 dimensional subspace. We write H(x) for the convex hull of the set of points x. For n from 0 to 3 the n-simplexes are:
 n n-simplex 0 point 1 line segment 2 triangle 3 tetrahedron
It goes on. We say that the n+1 points span the n-simplex. (Or at least that the simplex is the convex hull of the n+1 points.)

If V is the set of vertices of a simplex then H(V) is the simplex itself. For each of the 2n+1 subsets v of V the simplex H(v) is a subsimplex of H(V). That includes the −1-simplex which is the null set (Ø) which is spanned by Ø. The null simplex will not much concern us. The subsimplexes with the same number of vertices are generally considered together in proofs.

Within an n-simplex H(V) there is a notion of two sub simplexes opposite each other. If v1 ∩ v2 = Ø and v1 ∪ v2 = V then H(v1) and H(v2) are opposite sub-simplexes. For instance in a triangle there is the side opposite a vertex and vice-versa. In a tetrahedron there is a face opposite a vertex, an edge opposite another edge.

There is a definition something like:
For a given n-simplex and pair, B and C, of opposite sub-simplexes, the set of rays starting in B and passing thru C intersect a large concentric sphere in a set whose measure is the ‘angle’ at B. (This is the real definition.)