# Stable points of unit ball in Orlicz spaces

Commentationes Mathematicae Universitatis Carolinae (1991)

- Volume: 32, Issue: 3, page 501-515
- ISSN: 0010-2628

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topWisła, Marek. "Stable points of unit ball in Orlicz spaces." Commentationes Mathematicae Universitatis Carolinae 32.3 (1991): 501-515. <http://eudml.org/doc/247274>.

@article{Wisła1991,

abstract = {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 \lbrace (x,y):\frac\{1\}\{2\}(x+y)=z\rbrace $ 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 $S(L^\{\varphi \}(\mu ))$.},

author = {Wisła, Marek},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {stable point; stable unit ball; extreme point; Orlicz space; extreme point; stability of the unit ball in Orlicz spaces, endowed with the Luxemburg norm; lower-semicontinuous; stable points of norm},

language = {eng},

number = {3},

pages = {501-515},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Stable points of unit ball in Orlicz spaces},

url = {http://eudml.org/doc/247274},

volume = {32},

year = {1991},

}

TY - JOUR

AU - Wisła, Marek

TI - Stable points of unit ball in Orlicz spaces

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 1991

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 32

IS - 3

SP - 501

EP - 515

AB - 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 \lbrace (x,y):\frac{1}{2}(x+y)=z\rbrace $ 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 $S(L^{\varphi }(\mu ))$.

LA - eng

KW - stable point; stable unit ball; extreme point; Orlicz space; extreme point; stability of the unit ball in Orlicz spaces, endowed with the Luxemburg norm; lower-semicontinuous; stable points of norm

UR - http://eudml.org/doc/247274

ER -

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