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 $\varphi$.

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Orlicz 空间中的直径二性质和 Radon-Nikodým 性质

发现巴拿赫函数格具有氡-尼科狄姆性质的一些充分必要条件。结果表明，在非原子 $\sigma$-有限测度空间 $(\Omega,\Sigma,\mu)$ 上的 Orlicz 空间 $L_\varphi$，不一定可分离，具有 Radon-Nikodým 性质，如果并且仅当 $\varphi$ 是无穷远的 $N$ 函数并满足适当的 $\Delta_2$ 条件时。对于 Orlicz 序列空间 $\ell_\varphi$，当且仅当 $\varphi$ 满足条件 $\Delta_2^0$ 时，它才具有 Radon-Nikodým 性质。第二部分研究了Banach空间单位球面的均匀$\ell_1^2$点与切片直径的关系。使用这些结果，可以快速证明 Orlicz 空间 $L_\varphi$ 只有在 $\varphi$ 是线性的情况下才具有 Daugavet 性质，所以当 $L_\varphi$ 与 $L_1$ 等距时。另一个后果是配备了由 $N$ 函数生成的 Orlicz 范数的 Orlicz 空间永远不会具有局部直径二性质，而众所周知，当配备 Luxemburg 范数时，它可能具有该性质。最后，证明了局部直径二性质、直径二性质、强直径二性质在$\varphi$上的适当条件下在函数和序列Orlicz空间上与Luxemburg范数是等价的。