v2007.09.13 - Convex Optimization

2.12. CONVEX POLYHEDRA 1312.12.3 Unit simplexA peculiar convex subset of the nonnegative orthant havinghalfspace-descriptionS ∆ = {s | s ≽ 0, 1 T s ≤ 1} ⊆ R n + (253)is a unique bounded convex polyhedron called unit simplex (Figure 40)having nonempty interior, n + 1 vertices, and dimension [46,2.2.4]dim S = n (254)The origin supplies one vertex while heads of the standard basis [149][247] {e i , i=1... n} in R n constitute those remaining; 2.42 thus itsvertex-description:S2.12.3.1 Simplex= conv {0, {e i , i=1... n}}= { [0 e 1 e 2 · · · e n ]a | a T 1 = 1, a ≽ 0 } (255)The unit simplex comes from a class of general polyhedra called simplex,having vertex-description: [64] [228] [277] [77]conv{x l ∈ R n } | l = 1... k+1, dim aff{x l } = k , n ≥ k (256)So defined, a simplex is a closed bounded convex set having possibly emptyinterior. Examples of simplices, by increasing affine dimension, are: a point,any line segment, any triangle and its relative interior, a general tetrahedron,polychoron, and so on.2.12.3.1.1 Definition. Simplicial cone.A polyhedral proper (2.7.2.2.1) cone K in R n is called simplicial iff Khas exactly n extreme directions; [17,II.A] equivalently, iff proper K hasexactly n linearly independent generators contained in any given set ofgenerators.△There are an infinite variety of simplicial cones in R n ; e.g., Figure 15,Figure 41, Figure 50. Any orthant is simplicial, as is any rotation thereof.2.42 In R 0 the unit simplex is the point at the origin, in R the unit simplex is the linesegment [0,1], in R 2 it is a triangle and its relative interior, in R 3 it is the convex hull ofa tetrahedron (Figure 40), in R 4 it is the convex hull of a pentatope [279], and so on.