A. J. Hoffman's theorem and metric projections in polyhedral spaces
A polyhedral space is a Banach space whose unit ball is the convex hull of a finite set. A polyhedral space \(X\) can be considered as \(\mathbb{R}^ n\) endowed with a polyhedral norm \(\|\cdot\|\); i.e. there exist \(E_ 1,\dots,E_ m\in\mathbb{R}^ n\) such that \(\| z\|=\max_{1\leq i\leq m} E_ i^ T z\) for \(z\in\mathbb{R}^ n\), where the superscript ``\(T\)'' denotes the transpose of column vectors. Let \(B\) be an \(r\times n\) matrix, \(c\in\mathbb{R}^ n\), and \(K=\{x\in\mathbb{R}^ n: Bx\leq c\}\) a convex polyhedral subset of \(\mathbb{R}^ n\). For any point \(z\in\mathbb{R}^ n\), an element \(k_ 0\in K\) satisfying \[ d(z,k_ 0)= d(z,K)\equiv\min\{\| z-x\|: x\in K\} \] is called a best approximation to \(z\) in \(K\) and the metric projection \(P_ K\) from \(X\) to \(K\) is defined by the equation \[ P_ K(z)= \bigl\{x\in K: \| z-x\|= d(z,K)\bigr\}. \] The Lipschitz continuity and the Hausdorff strong uniqueness of \(P_ K\) is derived in this paper, i.e. the following is proved: Theorem. The metric projection \(P_ K\) is uniformly Hausdorff strongly unique and consequently Lipschitz continuous, i.e. there exists a positive constant \(\beta\), depending only on \(B\) and \(X\) such that \(\| z-x\|\geq d(z,K)+\beta d(x,P_ K(z))\) for \(x\in K\), \(z\in X\), and consequently \[ H(P_ K(\omega), P_ K(z))=\max\Bigl\{\max_{y\in P_ K(\omega)} d(y,P_ K(z)),\;\max_{x\in P_ K(z)} d(x,P_ K(\omega))\Bigr\}\leq {2\over \beta}\|\omega-z\| \] for \(\omega,z\in X\).
scientific article; zbMATH DE number 465064