Hoffman’s Error Bounds and Uniform Lipschitz Continuity of Best l -Approximations

In a central paper on smoothness of best approximation in 1968 R. Holmes and B. Kripke proved among others that on Rn, endowed with the lp-norm, 1-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 lp-approximations. As a consequence, we reprove and prove, respectively, Lipschitz continuity of the strict best approximation sba, ps`. and of the natural best approximation nba, ps1..