Uniform Hausdorff strong uniqueness
Let X be a normed linear space, M a proximinal subspace of X, \(P_ M(x)\) the set of all best approximations to x from M and \(P_ M\) a metric projection. In this article the definition of Hausdorff strong uniqueness of the set \(P_ M(x)\) and the definition of uniform Hausdorff strong uniqueness of the metric projection \(P_ M\) for any normed space X are given. The set \(P_ M(x)\) is said to be Hausdorff strongly unique if \[ r(x)=\inf \{\frac{\| x-m\| -d(x,M)}{d(m,P_ M(x))}:\quad m\in M\setminus P_ M(x)\}>0. \] The metric projection \(P_ M\) is said to be uniformly Hausdorff strongly unique if \(r(M)=\inf \{r(x):\) \(x\in X\}>0\). \textit{W. Li} [J. Approximation Theory 56, No.2, 164-184 (1989; Zbl 0672.41026)] defined and studied Hausdorff strong uniqueness in space C(T). In this article a characterization of (uniform) Hausdorff strong uniqueness is obtained. When M is a hyperplane, \(P_ M\) is Hausdorff strongly unique for some \(x\in X\setminus M\) if and only if \(P_ M\) is uniformly Hausdorff strongly unique. In C[a,b], \(P_ M\) is uniformly Hausdorff strongly unique if and only if M is one dimensional and Chebyshev.
scientific article; zbMATH DE number 4127832
scientific article