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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.

##### MSC:
 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 46B40 Ordered normed spaces
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