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Separation of convex sets and best approximation in spaces with asymmetric norm. (English) Zbl 1082.41024

The aim of this paper is to show that some results in Banach spaces have their analogs in spaces with asymmetric (non-symmetric) seminorms. A space with asymmetric norm is a pair $\left(X,p\right)$, where $X$ is a real vector space and $p$ a positive sublinear functional on $X$; $p\left(x\right)\ne p\left(-x\right)$ in general. First part of the paper deals with the dual space of $X$, extension of bounded linear functionals on $X$ and separation of convex sets.

The Krein-Milman theorem is stated as: Let $\left(X,p\right)$ be an asymmetric normed space such that its topolology ${\tau }_{p}$ (generated by the base of open balls $B\left(x,r\right)=\left\{y\in X\mid p\left(y-x\right) is Hausdorff. Then any nonempty ${\tau }_{p}$-compact convex subset of $X$ agrees with the ${\tau }_{p}$-closed convex hull of its extreme points.

Further, some characterisations and duality results for best approximation by elements of convex sets in spaces with asymmetric seminorms are considered, generalizing classical results of e.g. V.N. Nikolski, A.Garkavi and I. Singer. In the last sections duality results for the distance to a cavern (i.e., a complement to an open bounded set) is obtained, generalising the result proved by C. Franchetti and I. Singer in [Boll. Un. Math. Ital. B(5), 17, 33–43 (1980; Zbl 0436.41020)].

##### MSC:
 41A65 Abstract approximation theory 46A16 Non-locally convex linear spaces 46S99 Nonclassical functional analysis 46A20 Duality theory of topological linear spaces 46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators