# Article

 Title: Stable points of unit ball in Orlicz spaces  (English) Author: Wisła, Marek Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 32 Issue: 3 Year: 1991 Pages: 501-515 . Category: math . Summary: 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 $S(L^{\varphi }(\mu ))$. Keyword: stable point Keyword: stable unit ball Keyword: extreme point Keyword: Orlicz space MSC: 46B20 MSC: 46E30 idZBL: Zbl 0770.46013 idMR: MR1159798 . Date available: 2009-01-08T17:46:36Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/116986 . Reference: [1] Clausing A., Papadopoulou S.: Stable convex sets and extreme operators.Math. Ann. 231 (1978), 193-200. MR 0467249 Reference: [2] Engelking R.: General Topology.Polish Scientific Publishers, Warsaw, 1977. Zbl 0684.54001, MR 0500780 Reference: [3] Grząślewicz R.: Finite dimensional Orlicz spaces.Bull. Acad. Polon. Sci.: Math. 33 (1985), 277-283. MR 0816376 Reference: [4] Lazar A.J.: Affine functions on simplexes and extreme operators.Israel J. Math. 5 (1967), 31-43. Zbl 0149.08703, MR 0211246 Reference: [5] Lima Å.: On continuous convex functions and split faces.Proc. London Math. Soc. 25 (1972), 27-40. Zbl 0236.46024, MR 0303243 Reference: [6] Luxemburg W.A.J.: Banach function spaces.Thesis, Delft, 1955. Zbl 0162.44701, MR 0072440 Reference: [7] Michael E.: Continuous selections I.Ann. of Math. (2) 63 (1956), 361-382. Zbl 0071.15902, MR 0077107 Reference: [8] Musielak J.: Orlicz spaces and modular spaces.Lecture Notes in Math. 1034, Springer Verlag, 1983. Zbl 0557.46020, MR 0724434 Reference: [9] O'Brien R.C.: On the openness of the barycentre map.Math. Ann. 223 (1976), 207-212. Zbl 0321.46004, MR 0420221 Reference: [10] Orlicz W.: Über eine gewisse Klasse von Räumen vom Typus B.Bull. Intern. Acad. Pol., série A, Kraków, 1932, 207-220. Zbl 0006.31503 Reference: [11] Papadopoulou S.: On the geometry of stable compact convex sets.Math. Ann. 229 (1977), 193-200. Zbl 0339.46001, MR 0450938 Reference: [12] Suarez-Granero A.: Stable unit balls in Orlicz spaces.Proc. Amer. Math. Soc. 109, 1 (1990), 97-104. Zbl 0722.46014, MR 1000154 Reference: [13] Vesterstrøm J.: On open maps, compact convex sets and operator algebras.J. London Math. Soc. 6 (1973), 289-297. MR 0315464 Reference: [14] Wisła M.: Extreme points and stable unit balls in Orlicz sequence spaces.Archiv der Math. 56 (1991), 482-490. MR 1100574 Reference: [15] Wisła M.: Stable unit balls in finite dimensional generalized Orlicz spaces.Proceedings of the Second Conference Function Spaces'', Poznań, 1989, Teubner Texte zur Mathematik, to appear. MR 1155158 Reference: [16] Wisła M.: Continuity of the identity embedding of Musielak-Orlizc sequence spaces.Proc. of the 14th Winter School on Abstract Analysis, Srní, 1986, Supp. ai Rendiconti del Circolo Mat. di Palermo 14 (1987), 427-437. MR 0920876 .

## Files

Files Size Format View
CommentatMathUnivCarolRetro_32-1991-3_12.pdf 270.4Kb application/pdf View/Open

Partner of