Daugavet and diameter two properties in Orlicz-Lorentz spaces

In this article, we study the diameter two properties (D2Ps), the diametral diameter two properties (diametral D2Ps), and the Daugavet property in Orlicz-Lorentz spaces equipped with the Luxemburg norm. First, we characterize the Radon-Nikod\'ym property of Orlicz-Lorentz spaces in full generality by considering all finite real-valued Orlicz functions. To show this, the fundamental functions of their K\"othe dual spaces defined by extended real-valued Orlicz functions are computed. We also show that if an Orlicz function does not satisfy the appropriate $\Delta_2$-condition, the Orlicz-Lorentz space and its order-continuous subspace have the strong diameter two property. Consequently, given that an Orlicz function is an N-function at infinity, the same condition characterizes the diameter two properties of Orlicz-Lorentz spaces as well as the octahedralities of their K\"othe dual spaces. The Orlicz-Lorentz function spaces with the Daugavet property and the diametral D2Ps are isometrically isomorphic to $L_1$ when the weight function is regular. In the process, we observe that every locally uniformly nonsquare point is not a $\Delta$-point. This fact provides another class of real Banach spaces without $\Delta$-points. As another application, it is shown that for Orlicz-Lorentz spaces equipped with the Luxemburg norm defined by an N-function at infinity, their K\"othe dual spaces do not have the local diameter two property, and so as other (diametral) diameter two properties and the Daugavet property.Comment: 19 page

Diameter two properties and the Radon-Nikod\'ym property in Orlicz spaces

Some necessary and sufficient conditions are found for Banach function lattices to have the Radon-Nikod\'ym 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-Nikod\'ym 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-Nikod\'ym 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 $\varphi$

Clinicopathological features of infiltrating lobular carcinomas comparing with infiltrating ductal carcinomas: a case control study

BACKGROUND: Infiltrating lobular carcinoma (ILC) is the second most common type of invasive breast cancers and it has been reported to have some unique biologic and epidemiologic characteristics. METHODS: Clinicopathological features of 95 patients with ILC, their relapse free survival (RFS) and overall survival (OS) were retrospectively investigated and compared with those of 3,621 patients with infiltrating ductal carcinoma-not otherwise specified (IDC-NOS) between January 1984 and December 2005. RESULTS: ILC constitutes 2.3% of all invasive breast cancers. There were no difference between the ILC and the IDC-NOS groups regarding age at diagnosis, tumor size, nodal status, and treatment modalities except hormone therapy. The ILC group showed more estrogen receptor expression, less HER-2 expression and higher bilaterality. RFS and OS of the ILC patients were similar to those of the IDC. IDC-NOS metastasized more frequently to the lung and bone, whereas, ILC to the bone and ovary. CONCLUSIONS: The incidence of ILC was relatively low in Korean breast cancer patients. Comparing to IDC-NOS ILC showed some different features such as higher estrogen receptor expression, less HER-2 expression, higher bilaterality and preferred metastatic sites of bone and ovary. Contralateral cancers and bone and ovary evaluation should be considered when monitoring ILC patients