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Stable unit balls in Orlicz spaces. (English) Zbl 0722.46014

The author proves that if \(L^{\phi}(\mu)\) is an Orlicz space and \(X\subset L^{\phi}(\mu)\) an ideal such that for each \(f\in X\setminus \{0\}\) the modular \(I_{\phi}(f/\| f\|)=1\), then the closed unit ball \(B_ X\) of X is stable, that is, the midpoint map \((x,y)\to (x+y)\) from \(B_ X\times B_ X\) into \(B_ X\) is open. Here the \(\Delta\) \({}_ 2\) condition is not assumed as it is in the classical case. Stable sets have been studied by A. Clausing and S. Papadopoulou [Math. Ann. 231, 193-203 (1978; Zbl 0349.46002)], R. Grzaslewicz [Bull. Polish. Acad. Sci. Math. 33, 277-283 (1985; Zbl 0597.46024)], and S. Papadopoulou [Math. Ann. 229, 193-200 (1977; Zbl 0339.46001)].

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B20 Geometry and structure of normed linear spaces
47L07 Convex sets and cones of operators
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