Abstract
Extreme points of the unit sphere S (L 1+L ∞ ) of LL 1+L ∞ under the classical norm used in the interpolation theory were characterized in [8] and [11], while extreme points of S(L 1 ∩ L ∞) under the classical norm were characterized in [7]. In this paper extreme points of the unit sphere of L 1+L ∞ and L 1 ∩ L ∞ under the “dual” norms are characterized. Moreover, all the extreme points in L 1 ∩ L ∞ and L 1+L ∞ (under both kinds of norms) are examined if they are exposed, H-points, or strongly exposed. Smooth points in both these spaces for both the norms are also characterized. Finally, it is proved that in general the spaces L p +L q and L p ∩ L q are not isometric to Orlicz spaces, provided that 1<p<q<+∞.
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