HOFFMANS ERROR-BOUNDS AND UNIFORM LIPSCHITZ CONTINUITY OF BEST L(P)-APPROXIMATIONS

In a central paper on smoothness of best approximation in 1968 R. Holm es and B. Kripke proved among others that on R-n, endowed with the iot a(p)-norm, 1 <p < infinity the metric projection onto a given linear s ubspace is Lipschitz continuous where the Lipschitz constant depended on the parameter p. Using Hoffman's Error Bounds as a principal tool w e prove uniform Lipschitz continuity of best iota(p)-approximations. A s a consequence, we reprove and prove, respectively, Lipschitz continu ity of the strict best approximation (sba,p = infinity) and of the nat ural best approximation (nba, p = 1). (C) 1997 Academic Press.