Error estimates and Lipschitz constants for best approximation in continuous function spaces

Abstract

We use a structural characterization of the metric projection P G ( f ), from the continuous function space to its one-dimensional subspace G , to derive a lower bound of the Hausdorff strong unicity constant (or weak sharp minimum constant) for P G and then show this lower bound can be attained. Then the exact value of Lipschitz constant for P G is computed. The process is a quantitative analysis based on the Gâteaux derivative of P G , a representation of local Lipschitz constants, the equivalence of local and global Lipschitz constants for lower semicontinuous mappings, and construction of functions.