Continuous selection from the sets of best and near-best approximation

Abstract

The paper studies approximation and structural geometric-topological properties of sets in normed and more general (asymmetric) spaces for which there exists a continuous selection for the best and near-best approximation operators. Sufficient conditions on the metric projection of sets which ensure the existence of a continuous selection for this projection are obtained, and the structural properties of such sets are determined. The existence of a continuous selection for the near-best approximation operator on a finite-dimensional space more general than a normed space is investigated. It is shown that the lower semicontinuity of the metric projection is sufficient for the existence of a continuous selection for the near-best approximation operator in the general case.