11.4 THE FACETS AND VERTICES OF x1D49F_EuclidMathOne-Bold_11n_000100

A point px1D49F_EuclidMathOne_10n_000100 lies on the kth facet (or surface) of the upper hull if it satisfies the equation

(11.16) c11e016

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 x1D49F_EuclidMathOne_10n_000100. Similarly, a point px1D49F_EuclidMathOne_10n_000100 lies on the kth facet of the lower hull if it satisfies the equation

(11.17) c11e017

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 px1D49F_EuclidMathOne_10n_000100 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 ...

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