Let (T,\(\Sigma\),\(\mu)\) be a non-atomic measure space, let \(\phi\) : \(R\to (0,\infty)\) be an Orlicz function, and for \(\Sigma\)-measurable real function f on T, let \(I_{\phi}(\nu f)=\int_{T}\phi (\nu f(t))d\mu (t)\). Then \(L^{\phi}(\mu)\) denotes \(\{\) \(f: I_{\phi}(\nu f)0\}\), and \(\| f\|_{\phi}=\{a>0:\) \(I_{\phi}(f/a)\leq 1\}\). The function \(\phi\) satisfies condition \(\Delta_ 2\) if there is a constant \(K>0\) such that \(\phi\) (2u)\(\leq K\phi (u)\) for \(u\in R\). If \(f_{\phi}(a)=\sup_{u>0}\{2\phi ((u+au))/(\phi (u)+\phi (au))\}\), then p(\(\phi)\) denotes sup\(\{\) \(a\in (0,1):\) \(f_{\phi}(a)c\}.\) In one of the main theorems of this paper the authors show that \(\epsilon_ 0(L^{\phi})=2\) if \(\phi\) does not satisfy \(\Delta_ 2\), and \(\epsilon_ 0(L^{\phi})=2(1-p(\phi))/(1+p(\phi))\) if \(\phi\) satisfies \(\Delta_ 2\). Corollaries of the main results indicate conditions under which \(L^{\phi}\) is uniformly convex or non-square as in results obtained by \textit{A. Kamińska}, Indagiones Math. 44, 27-36 (1982; Zbl 0489.46025) and \textit{S. Chen}, Chin. Ann. Math., Ser. A 6, 619-624 (1985; Zbl 0595.46029).