This is an announcement for the paper “Diameter two properties and the Radon-Nikodým property in Orlicz spaces” by Anna Kamińskahttps://arxiv.org/search/math?searchtype=author&query=Kami%C5%84ska%2C+A, Han Ju Leehttps://arxiv.org/search/math?searchtype=author&query=Lee%2C+H+J, Hyung-Joon Taghttps://arxiv.org/search/math?searchtype=author&query=Tag%2C+H.
Abstract: Some necessary and sufficient conditions are found for Banach function lattices to have the Radon-Nikodým property. Consequently it is shown that an Orlicz space $L_\varphi$ over a non-atomic $\sigma$-finite measure space $(\Omega, \Sigma,\mu)$, not necessarily separable, has the Radon-Nikodým property if and only if $\varphi$ is an $N$-function at infinity and satisfies the appropriate $\Delta_2$ condition. For an Orlicz sequence space $\ell_\varphi$, it has the Radon-Nikodým property if and only if $\varphi$ satisfies condition $\Delta_2^0$. In the second part the relationships between uniformly $\ell_1^2$ points of the unit sphere of a Banach space and the diameter of the slices are studied. Using these results, a quick proof is given that an Orlicz space $L_\varphi$ has the Daugavet property only if $\varphi$ is linear, so when $L_\varphi$ is isometric to $L_1$. The other consequence is that the Orlicz spaces equipped with the Orlicz norm generated by $N$-functions never have local diameter two property, while it is well-known that when equipped with the Luxemburg norm, it may have that property. Finally, it is shown that the local diameter two property, the diameter two property, the strong diameter two property are equivalent in function and sequence Orlicz spaces with the Luxemburg norm under appropriate conditions on $\varphi$.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/2003.00396