Extreme Points and Reaction

Given a set of points, one is often interested in identifying which points have been generated from reaction and which have been generated from mixing operations. Let us explain how these points can be classified. 

Suppose that a convex set of k achievable concentrations in R , X = Cj, C2,. .., Cjt), is given. This set is assumed to have been obtained Irom the operations of a hypothetical reactor network, where many reactors may be linked by recycle, bypass and mixing operations. Suppose we are given a particular concentration C X and are tasked to determine whether  

Or C is obtained from a combination of mixing operations that does not employ reaction. 

A small amount of reasoning will help to determine the originating point within the reactor network. 

Concentrations Cj, C2,. .., Cj. are members of the set that make up the convex hull of X, conv(X). Recall that points belonging to the convex hull boundary are the unique points of the set. It is clear that the extreme points of the region defined by conv(X) are sufficient to generate the entire set of concentrations that lie in the set X. This is given 

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