Geometry of bilinear forms on the plane with the octagonal norm

Abstract

Let ℝ²_o(w) be the plane with the octagonal norm with weight 0 < w, w ≠ 1, ||(x; y)||_o(w) = max {|x| + w|y|, |y| + w|x|}. In this paper we classify all extreme, exposed and smooth points of the closed unit balls of ℒ(²ℝ2_o(w)) and ℒ_s(²ℝ²_o(w)); where ℒ(²ℝ²_o(w)) is the space of bilinear forms on ℝ²_o(w); and ℒ_s(²ℝ²_o(w)) is the subspace of ℒ(²l_{∞, θ}) consisting of symmetric bilinear forms.