Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space
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 \(E\).
scientific article; zbMATH DE number 733128