Ambiguous loci of the nearest point mapping in Banach spaces
Let \(E\) be a Banach space and let \(2^ E\) be the family of all nonempty subsets of \(E\). For \(X\in 2^ E\) and \(z\in E\), put \(d_ x (z)= \inf\{ \| x-z\|\): \( x\in X\}\), and consider the minimization problem \(\min (z,X)\) which consists in proving the existence of at least one point \(x\in X\) for which \(\| x-z \|= d_ x(z)\). Any such \(x\in X\) is called a solution of \(\min (z,X)\), and any sequence \(\{x_ n\} \subset X\) satisfying \(\lim_{n\to +\infty} \| x_ n-z \|= d_ x(z)\) is called a minimizing sequence of \(\min(z, X)\). \(\min(z, X)\) is said to be well-posed if it has a unique solution \(x\), say, and every minimizing sequence converges to \(x\). The authors study the properties of the amibiguous locus \[ A(X)= \{z\in E:\;\min (z,X) \text{ is not well-posed}\}, \] with \(X\) is closed and bounded in a separable strictly convex Banach space, and \(X\) is compact in a finite dimensional Euclidean space.
scientific article; zbMATH DE number 493385
scientific article