Lesniewski, Andrzej; Rzezuchowski, Tadeusz The Demyanov metric for convex, bounded sets and existence of Lipschitzian selectors. (English) Zbl 1227.52002 J. Convex Anal. 18, No. 3, 737-747 (2011). Let \({\mathcal E}^k\) be the set of orthonormal sequences \(E=(e_1,\dots,e_j)\) in \(\mathbb{R}^d\) with \(j\geq k\) for \(1\leq k\leq d\). For \(u\in\mathbb{R}^d\) and a convex bounded nonempty subset \(A\subset \mathbb{R}^d\) denote \(A(u)=\{a\in A: <a,u>= \sup_{x\in A} <x,u>\}\) and by recurrence \(A(u_1,\dots,u_i):=A(u_1,\dots,u_{i-1})(u_i)\). The Demyanov metric \(\rho_D\) in the space \({\mathcal K}^d\) of convex bodies in \(\mathbb{R}^d\) can be defined as \(\rho_D(A,B) = \sup_{E\in {\mathcal E}^k}\, \rho_H(A(E),B(E))\) where \(\rho_H\) denotes the usual Hausdorff metric. The authors modify slightly the Demyanow metric such that an extension is possible to convex, bounded but not necessarily closed sets, and they discuss the existence of Lipschitzian selectors. Reviewer: Eike Hertel (Jena) Cited in 5 Documents MSC: 52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces) 54E35 Metric spaces, metrizability 54C65 Selections in general topology 58C06 Set-valued and function-space-valued mappings on manifolds Keywords:convex and bounded sets; Demyanov metric; Hausdorff metric; linear selectors PDF BibTeX XML Cite \textit{A. Lesniewski} and \textit{T. Rzezuchowski}, J. Convex Anal. 18, No. 3, 737--747 (2011; Zbl 1227.52002) Full Text: Link OpenURL