Title: Weak neighborhoods and the Daugavet property of the interpolation spaces $L^1 + L^\infty$ and $L^1 \cap L^\infty$

Authors: Maria D. Acosta and Anna Kaminska

Issue: Volume 57 (2008), Issue 1, 77-96

Abstract: We study geometric properties of the spaces $\Sigma = L^{1} + L^{\infty}$ and $\Delta = L^{1} \cap L^{\infty}$ with the usual interpolation norms $\| \cdotp \|_{\Sigma}$ and $\| \cdotp \|_{\Delta}$, and their "dual" norms $|\!|\!| \cdotp |\!|\!|_{\Sigma}$ and $|\!|\!| \cdotp |\!|\!|_{\Delta}$. We show that none of the spaces $(\Sigma, \| \cdotp \|_{\Sigma})$, $(\Sigma, |\!|\!| \cdotp |\!|\!|_{\Sigma})$ and $(\Delta, \| \cdotp \|_{\Delta})$ has the Daugavet property, although the diameter of every nonempty weakly open subset of their unit balls is always $2$. The unit ball of $(\Delta, |\!|\!| \cdotp |\!|\!|_{\Delta})$ contains slices of arbitrarily small diameter, although none of its elements is strongly exposed. (See a graphic rendition of this abstract in http://www.iumj.indiana.edu/oai/2008/57/3171/3171_abstract.xml.)