Porosity and compacta with dense ambiguous loci of metric projections. (English) Zbl 1158.41314

Summary: Let \(X\) be a separable strictly convex Banach space and \(\mathcal{K}(X)\) th set of all nonempty compact subsets of \(X\) endowed with the Hausdorff metric. Let \(M\subset\mathcal{K}(X)\) consist of those compacta \(K\) for which the set of all points of multivaluedness of the metric projection onto \(K\) is not dense in \(X\). We show that \(M\) is a \(\sigma\)-porous set. The same holds for a class of separable non-strictly convex Banach spaces including \(\mathcal{C}([0,1])\) and also for all (non-separable) strictly convex Banach spaces.


41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46B20 Geometry and structure of normed linear spaces
54E52 Baire category, Baire spaces


Banach space
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