Extreme points and stable unit balls in Orlicz sequence spaces (Q909190)

A convex set C of a real Hausdorff topological vector space X is called stable if the midpoint map \((x,y)\to (x+y)\) from \(C\times C\) into C is open (C is endowed with the topology induced from X). It is well known that the set ext C is closed provided C is stable. In the first part of this paper there are given conditions which characterize the closedness of the set ext \(B_{\phi}\) \((=\) the set of all extreme points of the closed unit ball of an Orlicz sequence space \(\ell^ p\) equipped with the Luxemburg norm) for a subclass of Orlicz sequence spaces containing an isometric copy of \(\ell^{\infty}.\) Then the main theorem is proved: The unit ball \(B_{\phi}\) of an Orlicz sequence space \(\ell^{\phi}\) is stable if and only if either \(\phi\) satisfies the condition \(\Delta_ 2\) or \(\ell^{\phi}\) is isometric to \(\ell^{\infty}\).