Separation of convex sets

Definition 2.1.11 (Separating hyperplane) Let Si and Sj be nonempty sets in 3fn. The hyperplane 

Let S be a nonempty closed convex set in 5Rn, and a vector y which does not belong to the set S. Then there exist a nonzero vector c and a scalar z such that 

Let A be anmxn matrix and c be an n vector. Exactly one of the following two systems has a solution  

Remark 1 Farkas theorem has been used extensively in the development of optimality conditions for linear and nonlinear optimization problems. 

Let S and S2 be nonempty disjoint convex sets in 9 Then, there exists a hyperplane 

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