The Geometry of the Space of Symmetric Bilinear Forms on ℝ 2 with Octagonal Norm

. Let d ∗ (1 , w ) 2 = R 2 with the octagonal norm of weight w . It is the two dimensional real predual of Lorentz sequence space. In this paper we classify the smooth points of the unit ball of the space of symmetric bilinear forms on d ∗ (1 , w ) 2 . We also show that the unit sphere of the space of symmetric bilinear forms on d ∗ (1 , w ) 2 is the disjoint union of the sets of smooth points, extreme points and the set A as follows: where the set A consists of ax 1 x 2 + by 1 y 2 + c ( x 1 y 2 + x 2 y 1 ) with ( a = b = 0 , c = ± 1 1+ w 2 ), ( a (cid:54) = b, ab ≥ 0 , c = 0), ( a = b, 0 < ac, 0 < | c | < | a | ), ( a (cid:54) = | c | , a = − b, 0 < ac, 0 < | c | ), ( a = 1 − w 1+ w , b = 0 , c = 1 1+ w ), ( a = 1+ w + w ( w 2 − 3) c 1+ w predual