The Representation and Continuity of a Generalized Metric Projection onto a Closed Hyperplane in Banach Spaces

Abstract

Let be a closed bounded convex subset of a real Banach space with as its interior and the Minkowski functional generated by the set . For a nonempty set in and , is called the generalized best approximation to from if for all . In this paper, we will give a distance formula under from a point to a closed hyperplane in determined by a nonzero continuous linear functional in and a real number α, a representation of the generalized metric projection onto , and investigate the continuity of this generalized metric projection, extending corresponding results for the case of norm.