Metric projections and polyhedral spaces. (English) Zbl 1133.46008
Let \(X\) be a Banach space and \(Y\) a proximinal subspace of \(X\). The metric projection \(P_Y\) onto \(Y\) is called Hausdorff lower semi-continuous at \(x_0\in X\) if for every \(\epsilon >0\) there exists \(\delta=\delta(\epsilon)>0\) such that \(B(z,\epsilon)\cap P_Y(x)\neq \emptyset\) for every \(z\in P_Y(x_0)\) and every \(x\in B(x_0,\delta) = \{y\in X : \| y-x_0\| <\delta\}\). The main result of this paper (Theorem 1.15) asserts that if \(Y\) is a proximinal subspace of finite codimension of a Banach space \(X\) such that \(Y^\perp\) is polyhedral, then the metric projection \(P_Y\) is Hausdorff lower semi-continuous on \(X\). This completes previous results of the author and G.Godefroy [Rev.Mat.Complut.14, No. 1, 105–125 (2001; Zbl 0993.46004)] and of the author [Colloq.Math.99, No. 2, 231–252 (2004; Zbl 1061.46010)] (this last paper is misquoted as published in Collectanea Mathematicae). The present paper also surveys the properties of the metric projection onto subspaces of finite codimension in Banach spaces.
Reviewer: Stefan Cobzaş (Cluj-Napoca)
MSC:
| 46B20 | Geometry and structure of normed linear spaces |
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
References:
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