v2009.01.01 - Convex Optimization

2.12. CONVEX POLYHEDRA 137
2.12.3 Unit simplex
A peculiar subset of the nonnegative orthant with halfspace-description
S ∆ = {s | s ≽ 0, 1 T s ≤ 1} ⊆ R n + (265)
is a unique bounded convex polyhedron called unit simplex (Figure 46)
having nonempty interior, n + 1 vertices, and dimension [53,2.2.4]
dim S = n (266)
The origin supplies one vertex while heads of the standard basis [176]
[287] {e i , i=1... n} in R n constitute those remaining; 2.49 thus its
vertex-description:
S
2.12.3.1 Simplex
= conv {0, {e i , i=1... n}}
= { [0 e 1 e 2 · · · e n ]a | a T 1 = 1, a ≽ 0 } (267)
The unit simplex comes from a class of general polyhedra called simplex,
having vertex-description: [82] [266] [324] [96]
conv{x l ∈ R n } | l = 1... k+1, dim aff{x l } = k , n ≥ k (268)
So defined, a simplex is a closed bounded convex set having possibly empty
interior. 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 proper polyhedral (2.7.2.2.1) cone K in R n is called simplicial iff K has
exactly n extreme directions; [19,II.A] equivalently, iff proper K has exactly
n linearly independent generators contained in any given set of generators.
△
simplicial cone ⇒ proper polyhedral cone
There are an infinite variety of simplicial cones in R n ; e.g., Figure 20,
Figure 47, Figure 56. Any orthant is simplicial, as is any rotation thereof.
2.49 In R 0 the unit simplex is the point at the origin, in R the unit simplex is the line
segment [0,1], in R 2 it is a triangle and its relative interior, in R 3 it is the convex hull of
a tetrahedron (Figure 46), in R 4 it is the convex hull of a pentatope [326], and so on.