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