On the duality operator of a convex cone |
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Authors: | Bit-Shun Tam |
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Institution: | Department of Mathematics Tamkang University Tamsui, Taipei 251, Taiwan 251, Republic of China |
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Abstract: | Let C be a convex set in Rn. For each y?C, the cone of C at y, denoted by cone(y, C), is the cone {α(x ? y): α ? 0 and x?C}. If K is a cone in Rn, we shall denote by its dual cone and by F(K) the lattice of faces of K. Then the duality operator of K is the mapping given by . Properties of the duality operator dK of a closed, pointed, full cone K have been studied before. In this paper, we study dK for a general cone K, especially in relation to dcone(y, K), where y?K. Our main result says that, for any closed cone K in Rn, the duality operator dK is injective (surjective) if and only if the duality operator dcone(y, K) is injective (surjective) for each vector y?K ~ K ∩ (? K)]. In the last part of the paper, we obtain some partial results on the problem of constructing a compact convex set C, which contains the zero vector, such that cone (0, C) is equal to a given cone. |
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