Lower CS-closed sets and functions
Let \(X\) be a separated locally convex space. A subset \(C\) of \(X\) is called to be CS-closed (= ideally convex) if the sum of every convergent convex series on \(C\) (that is a series of the form \(\sum_{n=1}^{\infty} \lambda_n x_n\) where \(x_n \in C\), \(\lambda_n \geq 0\), and \(\sum \lambda_n = 1)\) is an element of \(C\). Every CS-closed set is convex, any closed or open convex set is CS-closed. A subset \(C\) of \(X\) is said to be lower CS-closed if there are a Fréchet space \(Y\) and a CS-closed subset \(A\) of \(X \times Y\) such that \(C\) is the projection of \(A\) on \(X\). Finally, a function on \(X\) is said to be lower CS-closed if its epigraph is a lower CS-closed subset of \(X \times \mathbb{R}\). The autors show various stability properties (e.g. the denumerable product and intersection of lower CS-closed sets are lower CS-closed) of lower CS-closed sets and functions, characterize lower CS-closed functions as marginal functions of CS-closed functions, establish an open mapping and a closed graph theorem, and show that any real extended valued lower CS-closed function on a metrizable barrelled space is continuous on the interior of its domain. Using this results the authors generalize classical theorems of convex duality by replacing lower semicountinuous functions by lower CS-closed ones. Moreover, some methods of convex analysis can be systematized and extended.
scientific article; zbMATH DE number 1384928
scientific article