Abstract
A cone C in Rn is a set of points such that if x∈ C, then so is every nonnegative scalar multiple of x, i.e., if x∈C, then λx∈C for 0 ≤ λ∈R, x∈Rn (see Figure 4.1.a for C in R2). If we consider the set of points X = {x}, then the cone generated by X is C = {y|y = λx, 0 ≤ λ∈ R, x∈X}. And if 0∉ X and for each y(≠0)∈C there are unique x∈X with λ > 0 such that y = λx, then X is termed a base of C. Moreover, the point 0∈Rn is termed the vertex of a cone and is an element of every cone since Y = 0x = 0∈C. In general, C is “not a subspace” of Rn since the definition of C holds only for λ ≥ 0 and not all λ.
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© 1993 Springer Science+Business Media Dordrecht
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Panik, M.J. (1993). Convex Cones in Rn. In: Fundamentals of Convex Analysis. Theory and Decision Library, vol 24. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8124-0_4
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DOI: https://doi.org/10.1007/978-94-015-8124-0_4
Publisher Name: Springer, Dordrecht
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