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The Demyanov metric for convex, bounded sets and existence of Lipschitzian selectors. (English) Zbl 1227.52002

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)

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
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