Ambiguous loci of the farthest distance mapping from compact convex sets

F. De Blasi; J. Myjak

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Displaying similar documents to “Ambiguous loci of the farthest distance mapping from compact convex sets”

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Boundaries of convex sets.

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Ehrhard Behrends (2000)

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The paper begins with a self-contained and short development of Bárány’s theorems of Carathéodory and Helly type in finite-dimensional spaces together with some new variants. In the second half the possible generalizations of these results to arbitrary Banach spaces are investigated. The Carathéodory-Bárány theorem has a counterpart in arbitrary dimensions under suitable uniform compactness or uniform boundedness conditions. The proper generalization of the Helly-Bárány theorem reads...

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We show that on every nonseparable Banach space which has a fundamental system (e.gȯn every nonseparable weakly compactly generated space, in particular on every nonseparable Hilbert space) there is a convex continuous function $f$ such that the set of its Gâteaux differentiability points is not Borel. Thereby we answer a question of J. Rainwater (1990) and extend, in the same time, a former result of M. Talagrand (1979), who gave an example of such a function $f$ on $ℓ^{1} (𝔠)$ .

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A closed convex subset C of a Banach space X is called approximatively polyhedral if for each ε > 0 there is a polyhedral (= intersection of finitely many closed half-spaces) convex set P ⊂ X at Hausdorff distance < ε from C. We characterize approximatively polyhedral convex sets in Banach spaces and apply the characterization to show that a connected component of the space $C o n v_{} (X)$ of closed convex subsets of X endowed with the Hausdorff metric is separable if and only if contains a...