Lower semicontinuity concepts, continuous selections, and set valued metric projections (Q1599256)

Metric projections onto finite dimensional subspaces of a Banach space are always upper semicontinuous, but generally they are not lower semicontinuous. Lower semicontinuity guarantees other basic properties of the metric projection. For example, such metric projections always admit a continuous selection. This paper distills the fundamental continuity concepts of metric projections that have such important implications. It investigates a number of semicontinuity properties, the relationship between them and their implications. For example, a geometric characterization is given of those normed linear spaces such that the metric projection onto each of its one dimensional subspaces has a continuous selection. Then the spaces \(C_O(T)\) and \(L_1(\mu)\) that have this property are determined.