LOCAL GEOMETRY OF L(1)BOOLEAN-AND-L(INFINITY) AND L(1)+L(INFINITY)

Extreme points of the unit sphere S(L(1) + L(infinity)) of L(1) + L(in finity) under the classical norm used in the interpolation theory were characterized in [8] and [11], while extreme points of S(L(1) boolean AND L(infinity)) under the classical norm were characterized in [7]. In this paper extreme points of the unit sphere of L(1) + L(infinity) and L(1) boolean AND L(infinity) under the ''dual'' norms are characte rized. Moreover, all the extreme points in L(1) boolean AND L(infinity ) and L(1) + L(infinity) (under both kinds of norms) are examined if t hey are exposed, II-points, or strongly exposed. Smooth points in both these spaces for both the norms are also characterized. Finally, it i s proved that in general the spaces L(p) + L(q) and L(p) boolean AND L (q) are not isometric to Orlicz spaces, provided that 1 < p < q < + in finity.