Characterizations for solidness of dual cones with applications (Q656833)

In the first part of the paper, based on a result due to \textit{G. Isac} [Topological methods in complementarity theory. Nonconvex Optimization and Its Applications. 41. Dordrecht: Kluwer Academic Publishers (2000; Zbl 0954.90056)], the author obtains several characterizations for solidness of dual cones. One of them is given in the following result: If \(K\) is a pointed convex cone in a normed space then its dual is a solid cone if and only if \(0\notin \overline{\text{co}}(\{x\in K:\|x\| = 1\})\). It is well-known that a pointed closed convex cone \(K\) in a reflexive Banach space has a weakly compact base if \(\text{int} K^* \neq \emptyset\). In this paper the following converse is established: A Banach space \(X\) is reflexive whenever there exists a solid pointed closed convex cone in \(X\) which has a weakly compact base. In the last part of the paper the author obtains, in reflexive Banach spaces, a global solvability result for linear complementarity problems, which generalizes an earlier result established by \textit{S. Karamardian} [J. Optimization Theory Appl. 19, 227--232 (1976; Zbl 0307.49010)] in Euclidean spaces.