THE UNIT BALL OF BILINEAR FORMS ON R² WITH A ROTATED SUPREMUM NORM

Let 0 ≤ θ <^π2^andl2∞,θ be the plane with the rotated supremum norm { } ∥(x, y)∥_∞,θ = max |(cosθ)x + (sinθ)y|, |(sinθ)x − (cosθ)y|. We devote to the description of the sets of extreme, exposed and smooth points of the closed unit balls of L(² l∞,θ^{2) and L} s(² l∞,θ^{2), where L(2} l∞,θ^{2) is} the space of bilinear forms on l∞,θ^{2,and L}s(² l∞,θ^{2) is the subspace of L(2} l∞,θ²⁾ consisting of symmetric bilinear forms. Let F = L(² l∞,θ^{2) or L}s(² l∞,θ²). First we classify the extreme and exposed points of the closed unit ball of F. We also show that every extreme point of the closed unit ball of F is exposed. It is shown that ext B_Ls(² l∞,θ²⁾ = ext B_L(² l∞,θ²⁾ ∩L_s (² l∞,θ^{2) and expB}L_s(² l∞,θ²⁾ = exp B_L(2 l∞,θ²⁾ ∩ L_s (² l∞,θ²). We classify the smooth points of the closed unit ball of F. It is shown that sm B_L(2 l∞,θ²⁾ ∩L_s (² l∞,θ^{2)⊊ smB}L_s(² l∞,θ²⁾. As corol-lary we extend the results of [18, 35].