## Ryszard Grz\text {\velkypolsky \char 161}\'{s}lewicz, Witold Seredy\'{n}ski

*Stability of positive part of unit ball in Orlicz spaces *

Comment.Math.Univ.Carolinae 46,3 (2005) 413-424. **Abstract:**The aim of this paper is to investigate the stability of the positive part of the unit ball in Orlicz spaces, endowed with the Luxemburg norm. The convex set $Q$ in a topological vector space is stable if the midpoint map $\Phi \colon Q\times Q\rightarrow Q$, $\Phi (x,y) =(x+y)/2$ is open with respect to the inherited topology in $Q$. The main theorem is established: In the Orlicz space ${L^\varphi (\mu )}$ the stability of the positive part of the unit ball is equivalent to the stability of the unit ball.

**Keywords:** stable convex set

**AMS Subject Classification:** Primary 52Axx, 46Axx,46Cxx

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