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Let X be a (real) vector space (e.g. ∝n or C(I)). A linear combination λ1 + λ2 x 2 + … + λx r x r of the vectors x 1, x 2, …, x r in X is called a convex combination of these vectors if the real numbers λ1 satisfy (Ai)λ i ≥ 0, and Σr/i≡ 1. The convex hull co E of a set E ⊂ X is the set of all convex combinations of finite sets of points in E. In particular, co x, yis the straight-line segment [x, y] joining the points x and y. A set E ⊂ X is convex if [x, y] ⊂ E whenever x,y ⊂E. (Equivalently, by an induction on the number of points, E is convex iff E = co E.)
KeywordsMathematical Program Extreme Point Convex Cone Convex Combination Linear Variety
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- Schaefer, H.H. (1966), Topological Vector Spaces, Macmillan, New York. (The separation theorem (2.2.3) is proved in Section II.9.)Google Scholar
- Valentine, F.A. (1964, 1976), Convex Sets, McGraw-Hill; Krieger. (See Parts I and II for finite-dimensional convexity and separation theorems.)Google Scholar