Projections onto closed convex sets in Hilbert spaces
Let \(X\) be a real Hilbert space. A set \(C\) in \(X\) is a Chebyshev set if for each point \(x\) in \(X\), there is a unique point of \(C\) that is nearest to \(x\); that is, there is a point \(q\) in \(C\) such that \(\| x- q\| <\| x-y\| \) for all \(y \in {C \setminus \{q\}}\). A mapping \(N:X\rightarrow C\) which takes each point \(x\in X\) to its nearest point \(q\) in \(C\) is called the metric projection. In this paper, the authors give necessary and sufficient algebraic conditions for a mapping \(F : X \rightarrow X\) with a closed image set to be the metric projection mapping onto a closed convex set. They provide examples that illustrate the necessity of each of the conditions. The characterizations given in the paper generalize several known results related to projections onto closed convex sets.
scientific article; zbMATH DE number 6806130
scientific article