Displaying similar documents to “Boundaries of convex sets.”

Complex Banach spaces with Valdivia dual unit ball.

Ondrej F. K. Kalenda (2005)

Extracta Mathematicae

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We study the classes of complex Banach spaces with Valdivia dual unit ball. We give complex analogues of several theorems on real spaces. Further we study relationship of these complex Banach spaces with their real versions and that of real Banach spaces and their complexification. We also formulate several open problems.

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 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 ) .

Ambiguous loci of the farthest distance mapping from compact convex sets

F. De Blasi, J. Myjak (1995)

Studia Mathematica

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Let be a strictly convex separable Banach space of dimension at least 2. Let K() be the space of all nonempty compact convex subsets of endowed with the Hausdorff distance. Denote by K 0 the set of all X ∈ K() such that the farthest distance mapping a ↦ M X ( a ) is multivalued on a dense subset of . It is proved that K 0 is a residual dense subset of K().