Abstract
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.)
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References
Ben-Israel, A. (1969), Linear equations and inequalities in finite dimensional, real or complex, vector spaces: a unified theory, J. Math. Anal. Appl., 27, 367–389. (For the counter example in 2.2)
Schaefer, H.H. (1966), Topological Vector Spaces, Macmillan, New York. (The separation theorem (2.2.3) is proved in Section II.9.)
Valentine, F.A. (1964, 1976), Convex Sets, McGraw-Hill; Krieger. (See Parts I and II for finite-dimensional convexity and separation theorems.)
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© 1978 B. D. Craven
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Craven, B.D. (1978). Mathematical techniques. In: Mathematical Programming and Control Theory. Chapman and Hall Mathematics Series. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5796-1_2
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DOI: https://doi.org/10.1007/978-94-009-5796-1_2
Publisher Name: Springer, Dordrecht
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