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Best approximation in a class of normed spaces with star-shaped cone. (English) Zbl 1098.41036
Let \(X\) be a real linear space. A cone \(K\) in \(X\) is called a star-shaped cone if the convex kernel of \(K\) is non empty (i.e. kern \(K:=\{u\in K\mid (x\in K,\) \(0\leq \alpha \leq 1)\Rightarrow u+\alpha (x-u)\in K\}\neq \phi ).\)
The authors consider star-shaped cones \(K\subset X\) with the following properties: a) \(K=\bigcup \{K_{i}:i\in I\},\) where \(I\) is an arbitrary index set; b) \(K_{i}(i\in I)\) is a closed convex pointed cone; c) intkern \(K\neq \phi \) and int \(K_{\ast }\neq \phi,\) where \(K_{\ast }=\bigcap \{K_{i}:i\in I\}.\) In this situation one considers the relation \(\leq _{K_{\ast }}\) and the downward sets and upward sets of \(X\) with respect to \( \leq _{K_{\ast}}.\) Also the space \(X\) is normed with the norm \(\left\| \cdot \right\| _{\ast },\) where \(\left\| x\right\| _{\ast }=\inf \{\lambda >0:x\leq _{K_{\ast }}\lambda {1},-x \leq _{K\ast }\lambda { 1} \}\) \((1\in\) int \(K_{\ast}\) is a fixed element), \(x\in X.\)
Then the best approximation properties of the downward and upward sets in \((X,\left\| \cdot \right\| _{\ast })\) are studied. Also the authors introduce the notion of relatively downward hull and relatively upward hull for arbitrary sets in \(X\) and study some of their properties. Finally, the obtained results are used for the examination of the best approximation by an arbitrary closed set.

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46B40 Ordered normed spaces
Full Text: DOI
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