Geometry of bilinear forms on the plane with hexagonal norms

Let 0 < w₁, w₂ < 1. We denote by R²_h(w1,w2)the plane with the hexagonal norm ∥(x, y)∥_h(w1,w2)= max { |y|, w₁ |x| + w₂ |y|}. We denote by R²_{h′ (w1,w2)} the plane with the hexagonal norm ∥(x, y)∥_{h′ (w1,w2)}= max {|x|, w₁ |x| + w₂ |y|}. In this paper, we classify the extreme bilinear forms of the unit balls of L(²X) and L_s(²X), where X = R²_h(w1,w2)or R²_h′(w1,w2). From this, we induce that ext B_Ls(²X) = ext B_L(²X) ∩ L_s(²X). We show that every extreme bilinear forms on that spaces is exposed.