Uniform Lipschitz continuity of best \(l_p\)-approximations by polyhedral sets (Q1276375)

Let \(\mathbb{R}^n\) be endowed with the \(p\)-norm, \(1<p<\infty\), and let \(K\) denote a polyhedral subset of \(\mathbb{R}^n\). The authors prove that the metric projection \(\Pi_{K,p}\) onto \(K\) of \(\mathbb{R}^n\) is uniformly Lipschitz continuous with respect to \(p\). As a consequence it follows that the strict best approximation and the natural best approximation are Lipschitz continuous selections for the metric projections \(\Pi_{K,\infty}\), and \(\Pi_{K, 1}\), respectively. The paper extends an analogous result on linear subspaces obtained by \textit{H. Berens}, \textit{M. Finzel}, \textit{W. Li} and \textit{Y. Xu} [J. Math. Anal. Appl. 213, No. 1, 183-201 (1997; Zbl 0891.41016)].