Stable points of unit ball in Orlicz spaces

Wisła, Marek

Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 3, pp. 501-515.

Résumé

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 $S(L^{\varphi }(\mu ))$.

Détail

Classification : 46B20, 46E30
Keywords: stable point; stable unit ball; extreme point; Orlicz space