Convex sets without diametral pairs

• Autores: Libor Veselý
• Localización: Extracta mathematicae, ISSN-e 0213-8743, Vol. 24, Nº 3, 2009, págs. 271-280
• Idioma: inglés
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• Resumen

  • Let X be an in¯nite dimensional normed linear space. It is not di±cult to see that arbitrarily near (in the Hausdor® 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 re°exive Banach space or X is a re°exive Banach space in which every weakly closed set contained in the unit sphere SX has empty relative interior in SX.