Weak neighborhoods and the Daugavet property of the interpolation spaces L1 + L and L1 ∪ L


We study geometric properties of the spaces Σ = L 1 + L∞ and Δ = L1 ∪L∞ with the usual interpolation norms || · ||Σ and || · ||Δ, and their "dual" norms ||| · ||| Σ and ||| · |||Δ. We show that neither of the spaces (Σ, || · ||Σ), (Σ, ||| · |||Σ) and (Δ, || · ||δ ) has the Daugavet property, although the diameter of every nonempty weakly open subset of their unit balls is always 2. The unit ball of (Δ, ||| · |||Δ) contains slices of arbitrarily small diameter, although none of its elements is strongly exposed. Indiana University Mathematics Journal ©.

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Indiana University Mathematics Journal