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 Ls(2l2∞ ,θ) is the subspace of L(2l2∞ ,θ) consisting of symmetric bilinear forms. Let F = L(2l2∞ ,θ) or Ls(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 BLs(2l2∞ ,θ) = extBL(2l2∞ ,θ) ∩Ls(2l2∞ ,θ) and exp BLs(2l2∞ ,θ) = exp BL(2l2∞ ,θ) ∩ Ls(2l2∞ ,θ). We classify the smooth points of the closed unit ball of F. It is shown that sm BL(2l2∞ ,θ)∩Ls(2l2∞ ,θ) ⊊ sm BLs(2l2∞ ,θ). As a corollary, we extend the results of [18, 35].