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Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space. (English) Zbl 0818.54026
The paper contains two main theorems and several auxiliary results. Let $$E$$ denote a real Banach space with norm $$\|\cdot \|$$ and $$S(E)$$ the family of all nonempty, compact, starshaped subsets of $$E$$ which is a complete metric space when endowed with the Hausdorff metric. Let $$\pi_ X$$ be the metric projection onto $$X$$, i.e. the mapping which associates to each $$a\in E$$ the set of all points in $$X$$ closest to $$a$$ and let $$A(X)$$ denote the ambiguous locus of $$X$$, i.e. $$A(X)= \{a\in E$$: $$\text{diam } \pi_ X (a)> 0\}$$. In a complete metric space $$M$$ the complement of any set of the first Baire category is called a residual subset of $$M$$ and its elements are called typical elements of $$M$$. A subset $$X$$ of a metric space $$M$$ is everywhere uncountable if for every $$a\in M$$ and $$r>0$$ the set $$X\cap B_ M (a,r)$$ is nonempty and uncountable. A set $$X$$ is starshaped with respect to a point $$u\in X$$ if the closed segment $$[u,x]$$ is contained in $$X$$ for every $$x\in X$$ and $$\ker X$$ is the set of all points with respect to which $$X$$ is starshaped. The authors prove that a set $$A$$ which denotes the set of all $$X\in S(E)$$ with ambiguous locus $$A(X)$$ everywhere uncountable in a strictly convex separable Banach space $$E$$ with $$\dim E\geq 2$$, is a residual subset of $$S(E)$$. They also prove that a typical element of $$S(E)$$ has a kernel consisting of a single point and a set of directions dense in the unit sphere of $$E$$.

##### MSC:
 54E52 Baire category, Baire spaces 52A30 Variants of convex sets (star-shaped, ($$m, n$$)-convex, etc.)
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