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

Kolář, Jan

Acta Univ. Carol., Math. Phys. 39, No. 1-2, 119-125 (1998).

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.

MSC:

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

Keywords:

Banach space