Abstract
Let A be a convex subset of ℝn. A point x ∈ A is an extreme point of A if and only if there exists no open segment contained in A and containing x, that is, if and only if there is no way to express x as a convex combination x = λy + (1 - λ)z such that y ∈A, z ∈ A and 0 < λ < 1, except by taking x = y = z. Easy computations show that an equivalent characterization for an extreme point x of A is that the relations x = (1/2)y + (l/2)z together with y ∈ A, z ∈ A imply x = y = z. Obviously, if A = a, then a is an extreme point of A.
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© 2001 Springer-Verlag Berlin Heidelberg
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Florenzano, M., Le Van, C. (2001). Extremal Structure of Convex Sets. In: Finite Dimensional Convexity and Optimization. Studies in Economic Theory, vol 13. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56522-9_3
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DOI: https://doi.org/10.1007/978-3-642-56522-9_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-62570-1
Online ISBN: 978-3-642-56522-9
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