11.4 THE FACETS AND VERTICES OF

A point p ∈ lies on the kth facet (or surface) of the upper hull if it satisfies the equation

(11.16)

This facet is of dimension 2 (i.e., n − 1). We can generalize by saying that multiplying the point p by a matrix of rank 1 results in the set of points that lies on a facet of dimension 1 less than n, the dimension of . Similarly, a point p ∈ lies on the kth facet of the lower hull if it satisfies the equation

(11.17)

We can extend the above argument and find all the points that satisfy two upper hull boundary conditions. Let us choose the two boundary conditions Ψ₁ and Ψ₂. Point p ∈ lies on the 1-2 facet of the upper hull when it satisfies the equation

(11.18)

This facet is of dimension 1 (i.e., n − 2) since Ψ₁ ≠ Ψ₂ by choice, which produces a matrix of rank 2. Since this facet is of dimension 1, it is actually a straight line describing the intersection of face 1 with face 2 of . This is an edge of the cubic volume.

Similarly, a domain point p ∈ lies on the 1-2 facet of the lower hull satisfies the equation

(11.19)

This facet is of dimension 1 (i.e., n − 2) since Λ₁ ≠ Λ₂ by choice. It is also possible to find the i–jth facets of that result due to the intersection of the upper and lower hulls by picking a Ψ_i and a Λ_j in the above constructions. The above procedure could be extended to construct 3 × 3 matrices of rank 3 to obtain the vertices of .