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Stability of positive part of unit ball in Orlicz spaces. (English) Zbl 1121.52001

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 \(\Phi \: Q\times Q\rightarrow Q\), \(\Phi (x,y) =(x+y)/2\) 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.

MSC:

52Axx General convexity
46Axx Topological linear spaces and related structures
46Cxx Inner product spaces and their generalizations, Hilbert spaces
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