Local completeness and drop theorem
Let \(E\) be a Banach space with closed unit ball \(B\) and \(x\in E\setminus B\). The ``drop'' generated by \(x\) and \(B\) is the set \(D(x,B)=\operatorname {co}(B\cup\{x\})\). Danes drop theorem asserts that for every nonempty closed set \(A\subset E\) situated at a positive distance from \(B\) there exists \(a\in A\) such that \(D(a,B)\cap A = \{a\}\) [\textit{J. Daneš}, Boll. Un. Mat. Ital. 6, 369--372 (1972; Zbl 0236.47053) and Commentat. Math. Univ. Carol. 26, 443--454 (1985; Zbl 0656.47050)]. The drop property turned to be a very important and useful geometric tool, equivalent to Ekeland's variational principle [see \textit{J.-P. Penot}, Nonlinear Analysis 10, 813--822 (1986; Zbl 0612.49011)]. The aim of the present paper is to extend this property to locally convex spaces. The main innovation of the author consists in the use of the notions of local closedness and local completeness of subsets of a locally convex space \(E\), defined in terms of the Minkowski gauge of some discs in \(E\). These notions are weaker than those of sequential closedness and sequential completeness, used by \textit{L. Cheng, Y. Zhou} and \textit{F. Zhang} [Proc. Am. Math. Soc 124, No. 12, 3699--3702 (1996; Zbl 0863.46003)], to prove a drop theorem in locally convex spaces.
scientific article; zbMATH DE number 1744276
scientific article