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A selection theorem of Helly type and its applications. (English) Zbl 0847.52004

The authors prove a Helly-type abstract selection theorem for set-valued mappings with convex compact values in a normed space. In the first theorem under the hypothesis in which \(\Phi : D \to cc (Y)\) is a set valued mapping from a set \(X\) to a normed space \((Y, |, |)\) and \(W\) an \(\ell\)-dimensional subspace of all mappings from \(X\) to \( Y\) and for each \(f \in W\), \(D\) is assumed to have enough points for \(f |_p = 0\) to imply \(f = 0\), the authors prove the existence of \(h \in W\) such that \(h(x) \in \Phi (x)\) for \(\forall x \in D \) or equivalently the existence of \(h \in W\) such that \(h(x) \in \Phi (x)\) for \(\forall x \in \widetilde D\), \(\widetilde D\) being any subset of \(D\) with the cardinality \(\ell + 1\). In the corollaries the necessary and sufficient conditions are given for set valued mappings to have constant, affine, linear and polynomial selection. The second theorem is a generalization of the sandwich theorem due to K. Nikodem and S. Wasowicz. It is about the separation of two real functions defined and a subset of \(R^n\) by an affine one. And finally for affine functions a Hyers-Ulam type stability theorem is given.

MSC:

52A35 Helly-type theorems and geometric transversal theory
52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
52A41 Convex functions and convex programs in convex geometry
26B25 Convexity of real functions of several variables, generalizations
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