Two remarks on diameter 2 properties; pp. 2–7Full article in PDF format | doi: 10.3176/proc.2014.1.02
A Banach space is said to have the diameter 2 property if the diameter of every nonempty relatively weakly open subset of its unit ball equals 2. In a paper by Abrahamsen, Lima, and Nygaard (Remarks on diameter 2 properties. J. Conv. Anal., 2013, 20, 439–452), the strong diameter 2 property is introduced and studied. This is the property that the diameter of every convex combination of slices of its unit ball equals 2. It is known that the diameter 2 property is stable by taking ℓp-sums for 1 ≤ p ≤ ∞. We show the absence of the strong diameter 2 property on ℓp-sums of Banach spaces when 1 < p < ∞. This confirms the conjecture of Abrahamsen, Lima, and Nygaard that the diameter 2 property and the strong diameter 2 property are different. We also show that the strong diameter 2 property carries over to the whole space from a non-zero M-ideal.
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