Characterizations of some remarkable classes of cones
A number of different classes of cones whose definitions make sense in topological vector spaces are collected in this paper from several references, and some relations among them are proved. In the first part, the properties of being nonexplosive, pseudonuclear, nuclear and strongly exposed are reduced to geometric properties of the vertex of the cone, in locally convex setting. The second part deals with normed spaces. The list of properties of cones and connections among them is increased and the main result of the paper is proved. One of the multiple formulations of this result (Theorem 3) asserts that given a convex cone \(C\) with pointed closure in a Banach space, the following properties are equivalent: (i) \(C\) is nonexplosive (that is, the neighborhoods of zero for the topologies on \(C\) induced by norm and weak topologies coincide). (ii) \(C\) satisfies the angle property, as defined by \textit{Cesari} and \textit{Suryanarayana} [Trans. Amer. Math. Soc. 244, 37-65 (1978; Zbl 0393.49004)] or, using another terminology, \(C\) is a nuclear cone. (It is also shown that the assumption of completeness cannot be dropped.) A similar equivalence had been proved for reflexive Banach spaces by \textit{K. L. Chew} [J. Optim. Theory Appl. 44, 1-53 (1984; Zbl 0529.46023)].
scientific article; zbMATH DE number 1915750