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Keywords:
stable convex set
stable convex set
Summary:
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.
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