Mathematical techniques

Part of the Chapman and Hall Mathematics Series book series (CHMS)


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 i ≥ 0, and Σr/i≡ 1. The convex hull co E of a set EX 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 EX is convex if [x, y] ⊂ E whenever x,yE. (Equivalently, by an induction on the number of points, E is convex iff E = co E.)


Mathematical Program Extreme Point Convex Cone Convex Combination Linear Variety 
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  1. 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)CrossRefGoogle Scholar
  2. Schaefer, H.H. (1966), Topological Vector Spaces, Macmillan, New York. (The separation theorem (2.2.3) is proved in Section II.9.)Google Scholar
  3. Valentine, F.A. (1964, 1976), Convex Sets, McGraw-Hill; Krieger. (See Parts I and II for finite-dimensional convexity and separation theorems.)Google Scholar

Copyright information

© B. D. Craven 1978

Authors and Affiliations

  1. 1.University of MelbourneAustralia

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