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    Title: On the daulity operator of a convex cone
    Authors: Tam, Bit-shun
    Contributors: 淡江大學數學學系
    Date: 1985-01
    Issue Date: 2010-01-28 07:29:08 (UTC+8)
    Publisher: New York : Elsevier Inc.
    Abstract: Let C be a convex set in Rn. For each yϵC, the cone of C at y, denoted by cone(y, C), is the cone {α(x − y): α ⪖ 0 and xϵC}. If K is a cone in Rn, we shall denote by K∗ its dual cone and by F(K) the lattice of faces of K. Then the duality operator of K is the mapping View the MathML source given by View the MathML source. 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 yϵK. 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 yϵK ∼ [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.
    Relation: Linear Algebra and Its Applications 64, pp.33-56
    DOI: 10.1016/0024-3795(85)90265-4
    Appears in Collections:[數學學系暨研究所] 期刊論文

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