Convex Sets

Abstract

The theorems proved in this chapter provide the mathematical background for the theory of linear programming. Convex sets are introduced, not only to prove some of the theorems, but also to give the reader a geometrical picture of the algebraic processes involved in solving a linear programme. In the following definitions, Rⁿ denotes the vector space of all real n-vectors, Definition. A subset¹ C ⊂ Rⁿ is convex if for all² u_l u₂ ∈ C.

$$ \lambda {u_1} + \left( {1 - \lambda } \right){u_2} \in C\;for\;0 \leqslant \lambda \leqslant 1 $$

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