Properties of typical bounded closed convex sets in Hilbert space.
de Blasi, F.S., Zhivkov, N.V. (2005)
Abstract and Applied Analysis
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Displaying similar documents to “Ambiguous loci of the farthest distance mapping from compact convex sets”
Properties of typical bounded closed convex sets in Hilbert space.
Boundaries of convex sets.
M. Valdivia (1993)
Revista de la Real Academia de Ciencias Exactas Físicas y Naturales
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On Bárány's theorems of Carathéodory and Helly type
Ehrhard Behrends (2000)
Studia Mathematica
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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...
Convex functions with non-Borel set of Gâteaux differentiability points
Petr Holický, M. Šmídek, Luděk Zajíček (1998)
Commentationes Mathematicae Universitatis Carolinae
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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 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 on .
Fragmentability and compactness in C(K)-spaces
B. Cascales, G. Manjabacas, G. Vera (1998)
Studia Mathematica
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Let K be a compact Hausdorff space, the space of continuous functions on K endowed with the pointwise convergence topology, D ⊂ K a dense subset and the topology in C(K) of pointwise convergence on D. It is proved that when is Lindelöf the -compact subsets of C(K) are fragmented by the supremum norm of C(K). As a consequence we obtain some Namioka type results and apply them to prove that if K is separable and is Lindelöf, then K is metrizable if, and only if, there is a countable...
Supergeneric results and Gateâux differentiability of convex and Lipschitz functions on small sets
Luděk Zajíček (1997)
Acta Universitatis Carolinae. Mathematica et Physica
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A "hidden" characterization of approximatively polyhedral convex sets in Banach spaces
Taras Banakh, Ivan Hetman (2012)
Studia Mathematica
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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 of closed convex subsets of X endowed with the Hausdorff metric is separable if and only if contains a...