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A full characterization of stable unit balls in Orlicz spaces. (English) Zbl 0772.46012

A convex subset \(C\) of a topological vector space is called stable if the midpoint map \(\varphi_{1/2}: C\times C\to C\) is open, where \(\varphi_{1/2}(x,y)={1\over 2}(x+y)\). The author gives a full characterization of stable unit balls in Orlicz spaces.

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
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References:

[1] A. Clausing and S. Papadopoulou, Stable convex sets and extremal operators, Math. Ann. 231 (1977/78), no. 3, 193 – 203. · Zbl 0349.46002
[2] Antonio Suárez Granero, Stable unit balls in Orlicz spaces, Proc. Amer. Math. Soc. 109 (1990), no. 1, 97 – 104. · Zbl 0722.46014
[3] Julian Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983. · Zbl 0557.46020
[4] Susanna Papadopoulou, On the geometry of stable compact convex sets, Math. Ann. 229 (1977), no. 3, 193 – 200. · Zbl 0339.46001
[5] Marek Wisła, Extreme points and stable unit balls in Orlicz sequence spaces, Arch. Math. (Basel) 56 (1991), no. 5, 482 – 490. · Zbl 0694.46004
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