Abstract

If X is an infinite-dimensional uniform algebra, if X has the Daugavet property or if X is a proper M-embedded space, every relatively weakly open subset of the unit ball of the Banach space X is known to have diameter 2, i.e., X has the diameter 2 property. We prove that in these three cases even every finite convex combination of relatively weakly open subsets of the unit ball have diameter 2. Further, we identify new examples of spaces with the diameter 2 property outside the formerly known cases; in particular we observe that forming ℓp\ell_p-sums of diameter 2 spaces does not ruin diameter 2 structure.

Author information

Contact details are reproduced from the original publication and may be historical.

Olav Nygaard

Dept. of Mathematics, Agder University, Servicebox 422, 4604 Kristiansand, Norway

Olav.Nygaard@uia.no

Keywords

  • Diameter 2
  • slice
  • Daugavet property
  • M-embedded
  • uniform algebra

Mathematics Subject Classification

46B20, 46B22

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Suggested citation

T. Abrahamsen, V. Lima, O. Nygaard. “Remarks on Diameter 2 Properties.” Journal of Convex Analysis 20 (2013), No. 2, 439–452. https://doi.org/10.68381/jca20026

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.