The aim of this paper is to investigate stability of unit ball in Orlicz spaces, endowed with the Luxemburg norm, from the ``local'' point of view. Firstly, those points of the unit ball are characterized which are stable, i.e., at which the map $z\rightarrow \{(x,y):\frac{1}{2}(x+y)=z\}$ is lower-semicontinuous. Then the main theorem is established: An Orlicz space $L^{\varphi }(\mu )$ has stable unit ball if and only if either $L^{\varphi }(\mu )$ is finite dimensional or it is isometric to $L^{\infty }(\mu )$ or $\varphi $ satisfies the condition $\Delta _r$ or $\Delta _r^0$ (appropriate to the measure $\mu $ and the function $\varphi $) or $c(\varphi )<\infty , \varphi (c(\varphi ))<\infty $ and $\mu (T)<\infty $. Finally, it is proved that the set of all stable points of norm one is dense in the unit sphere $S(L^{\varphi }(\mu ))$.
@article{116986,
author = {Marek Wis\l a},
title = {Stable points of unit ball in Orlicz spaces},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {32},
year = {1991},
pages = {501-515},
zbl = {0770.46013},
mrnumber = {1159798},
language = {en},
url = {http://dml.mathdoc.fr/item/116986}
}
Wisła, Marek. Stable points of unit ball in Orlicz spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) pp. 501-515. http://gdmltest.u-ga.fr/item/116986/
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