On Relatively Solid Convex Cones in Real Linear Spaces

Vicente Novo () and Constantin Zălinescu ()
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Vicente Novo: Universidad Nacional de Educación a Distancia
Constantin Zălinescu: Iaşi Branch of Romanian Academy

Journal of Optimization Theory and Applications, 2021, vol. 188, issue 1, No 13, 277-290

Abstract: Abstract Having a convex cone K in an infinite-dimensional real linear space X, Adán and Novo stated (in J Optim Theory Appl 121:515–540, 2004) that the relative algebraic interior of K is nonempty if and only if the relative algebraic interior of the positive dual cone of K is nonempty. In this paper, we show that the direct implication is not true even if K is closed with respect to the finest locally convex topology $$\tau _{c}$$ τ c on X, while the reverse implication is not true if K is not $$\tau _{c}$$ τ c -closed. However, in the main result of this paper, we prove that the latter implication is true if the algebraic interior of the positive dual cone of K is nonempty; the general case remains an open problem. As a by-product, a result about separation of cones is obtained that improves Theorem 2.2 of the work mentioned above.

Keywords: Algebraic relative interior; Algebraic dual cone; Algebraic and vectorial closures; Topological dual cone; Convex core topology; 52A05; 06F20; 90C48 (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-020-01773-z

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