Uniform Lipschitz continuity of best lp-approximations by polyhedral sets

In this paper we prove that the metric projection Pi(K, p) onto a polyhedral subset K of R-n, endowed with the p-norm, is uniformly Lipschitz continuous with respect to p, 1 < p < infinity. As a consequence the strict best approximation and the natural best approximation are Lipschitz continuous selections for the metric projections Pi(K, infinity) and Pi(K, 1), respectively. This extends a recent analogous result in Berens et al. [J. Math. Anal. Appl. 213 (1997), 183-201] on linear subspaces. (C) 1998 Academic Press.