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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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[1] DOI: 10.1007/BF01891408 · Zbl 0682.41034 · doi:10.1007/BF01891408
[2] DOI: 10.1016/0021-9045(92)90117-7 · Zbl 0796.41026 · doi:10.1016/0021-9045(92)90117-7
[3] Deutch F., Best Approximation in Inner Product Spaces (2000)
[4] DOI: 10.1006/jath.1996.3082 · Zbl 0884.41019 · doi:10.1006/jath.1996.3082
[5] DOI: 10.1137/S1052623498337273 · Zbl 0957.41025 · doi:10.1137/S1052623498337273
[6] DOI: 10.1023/A:1015583411806 · Zbl 1013.90101 · doi:10.1023/A:1015583411806
[7] Mohebi H., J. Anal. Theory Appl. 22 pp 1–
[8] Mohebi H., J. Math. Operations Res. 31 pp 124– · Zbl 1278.90319 · doi:10.1287/moor.1050.0173
[9] DOI: 10.1006/jath.1996.3107 · Zbl 0892.41005 · doi:10.1006/jath.1996.3107
[10] Rubinov A.M., Abstract Convex Analysis and Global Optimization (2000) · Zbl 0985.90074
[11] DOI: 10.1023/B:JOGO.0000047914.22567.66 · Zbl 1080.90068 · doi:10.1023/B:JOGO.0000047914.22567.66
[12] DOI: 10.1080/02331930108844567 · Zbl 1007.26010 · doi:10.1080/02331930108844567
[13] DOI: 10.1006/jath.2000.3495 · Zbl 0967.41017 · doi:10.1006/jath.2000.3495
[14] Singer I., Abstract Convex Analysis (1997) · Zbl 0898.49001
[15] Singer I., Best approximation in Normed Linear Spaces by Elements of Linear Subspaces (1970) · Zbl 0197.38601
[16] DOI: 10.1081/NFA-200063880 · Zbl 1072.41019 · doi:10.1081/NFA-200063880
[17] DOI: 10.1016/j.jat.2005.04.004 · Zbl 1138.41304 · doi:10.1016/j.jat.2005.04.004
[18] Vlasov L.P., Math. Notes 2 pp 600– (1967)
[19] DOI: 10.1070/RM1973v028n06ABEH001624 · Zbl 0293.41031 · doi:10.1070/RM1973v028n06ABEH001624
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