## 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.

### 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

Banach space
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