In a central paper on smoothness of best approximation in 1968 R. Holmes and B. Kripke proved among others that on ‫ޒ‬ n , endowed with the l-norm, 1p-ϱ, p the metric projection onto a given linear subspace is Lipschitz continuous where the Lipschitz constant depended on the parameter p. Using Hoffman's Error Bounds as a principal tool we prove uniform Lipschitz continuity of best l-app proximations. As a consequence, we reprove and prove, respectively, Lipschitz Ž. continuity of the strict best approximation sba, p s ϱ and of the natural best Ž. approximation nba, p s 1 .

The main goal of this paper is to show the connection between optimization and best approximation when studying vector-valued functions defined on a finite set. For example, Hausdorff strong unicity for best approximation is shown to be equivalent to Abadie's constraint qualification for the associated convex quadratic feasibility problem. Academic Press

We say that a subset G of=G for any k_k orthogonal matrix Q. Let G be a rotation-invariant finite-dimensional subspace of C 0 (T, R k ) on a connected, locally compact, metric space T. We prove that G is a generalized Haar subspace if and only if P G ( f ) is strongly unique of order 2 whenever P G ( f ) is a singleton.1998 Academic Press