The unit ball of biliniar forms on R2 with a rotated supremum norm

Abstract

Let 0 ≤ θ < π 2 and l²_{∞ ,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(²l²_{∞ ,θ}) and Ls(²l²_{∞ ,θ}), where L(²l2∞ ,θ) is the space of bilinear forms on l²_{∞ ,θ}, and L_s(²l²_{∞ ,θ}) is the subspace of L(²l²_{∞ ,θ}) consisting of symmetric bilinear forms. Let F = L(²l2_{∞ ,θ}) or L_s(²l²_{∞ ,θ}). 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(2l2∞ ,θ)} = ext_{BL(2l2∞ ,θ)} ∩L_s(²l²_{∞ ,θ}) and exp B_{Ls(2l2∞ ,θ)} = exp B_{L(²l²∞ ,θ)} ∩ L_s(²l²_{∞ ,θ}). We classify the smooth points of the closed unit ball of F. It is shown that sm B_{L(2l2∞ ,θ)}∩L_s(²l²_{∞ ,θ}) ⊊ sm B_{Ls(²l²∞ ,θ)}. As a corollary, we extend the results of [18, 35].