On some convexity properties of Orlicz spaces of vector valued functions. (English) Zbl 0734.46020
Summary: A “stability” theorem that is a generalization of Theorem 6 in [D. J. Downing and B. Turett, Pac. J. Math. 104, No.2, 343-350 (1983; Zbl 0464.47036)] for the modulus of convexity of Banach spaces is given. Necessary and sufficient conditions for \(\delta_ L\Phi (a)>0\), where \(a\in (0,2]\), in Orlicz spaces \(L^{\Phi}(\mu,X)\) of vector valued functions are given. The convexity coefficient \(\epsilon_ 0(L^{\Phi}(\mu,X))\) is computed for these spaces. The equality \(\epsilon_ 0(L^{\Phi}(\mu,X))=\epsilon_ 0(X)\) for Orlicz-Bochner spaces generated by uniformly convex Orlicz functions satisfying the \(\Delta_ 2\)-condition is shown.
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46E40 | Spaces of vector- and operator-valued functions |
| 46B20 | Geometry and structure of normed linear spaces |
