Lower Semicontinuity Concepts, Continuous Selections, and Set Valued Metric Projections

A number of semicontinuity concepts and the relations between them are discussed. Characterizations are given for when the (set-valued) metric projection PM onto a proximinal subspace M of a normed linear space X is approximate lower semicontinuous or 2-lower semicontinuous. A geometric characterization is given of those normed linear spaces X such that the metric projection onto every one-dimensional subspace has a continuous C0(T) and L1(μ) that have this property are determined.