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Title: Convex sets without diametral pairs
Authors: Veselý, Libor
Keywords: Diametral pair;Bounded closed convex set;Hausdorff metric;Par diametral;Conjunto convexo cerrado acotado;Métrica de Hausdorff
Issue Date: 2009
Publisher: Universidad de Extremadura, Servicio de Publicaciones
Abstract: Let X be an infinite dimensional normed linear space. It is not difficult to see that arbitrarily near (in the Hausdorrff metric) to the unit ball of X there exists a nonempty closed convex set whose diameter is not attained. We show that such sets are dense in the metric space of all nonempty bounded closed convex subsets of X if and only if either X is not a reflexive Banach space or X is a reflexive Banach space in which every weakly closed set contained in the unit sphere Sx has empty relative interior in Sx.
ISSN: 0213-8743
Appears in Collections:Extracta Mathematicae Vol. 24, nº 3 (2009)

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