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On the size of the set of points where the metric projection exists. (English) Zbl 1067.46015

A closed convex set \(C\) in a reflexive, locally uniformly convex Banach space \(X\) admits best approximations to a dense \(G_{\delta}\) subset of \(X\) [E. Asplund, Isr. J. Math. 4, 213–216 (1966; Zbl 0143.34904)]. This work answers negatively a conjecture that if the hypothesis of the local uniform convexity is reduced to strict convexity then almost every \(x \in X\) has a best approximation in \(C\). A similar negative result is presented for farthest points and a conjecture that a property of differentials in reflexive spaces remains true for a non-reflexive space is shown to be false.

MSC:

46B20 Geometry and structure of normed linear spaces
41A50 Best approximation, Chebyshev systems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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References:

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