On the daulity operator of a convex cone

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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:	[Graduate Institute & Department of Mathematics] Journal Article

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