On suns and cosuns in finite dimensional normed real vector spaces. (English) Zbl 0592.41043

Given a finite-dimensional normed linear space X and a nonempty subset A of X, we define \(\bar x\in A\) to be a best coapproximation to \(x\in X\) if \(\| \bar x-x\| =\min \{\| a-x\|;a\in A\}\). If we borrow from the theory of best approximation the notions of sun, Chebyshev set and so on, we can define the akin notions of cosun, co-Chebyshev set, and so on. Here a detailed study of the following notions is accomplished: sets defined by \(E_ X(x,y)=\{z\in X;\| x-z\| <\| y-z\| \}\) in X (especially in \(\ell^ p_ 3)\); cosuns and related notions; suns and cosuns in two-dimensional spaces, by using dual norms.
This deep study is the author’s doctoral thesis; note that while two- dimensional spaces are quite special, three-dimensional spaces already contain practically all the pathologies of general normed spaces. For some related problems, see the survey paper by the reviewer [Rend. Semin. Mat. Fis. Milano 53, 131-148 (1983)].
Reviewer: P.L.Papini


41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A50 Best approximation, Chebyshev systems
46B99 Normed linear spaces and Banach spaces; Banach lattices
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