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

초록

Let 0 ≤ θ <π2andl2∞,θ be the plane with the rotated supremum norm { } ∥(x, y)∥<inf>∞,θ</inf> = 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∞,θ2) and L s(2 l∞,θ2), where L(2 l∞,θ2) is the space of bilinear forms on l∞,θ2,and Ls(2 l∞,θ2) is the subspace of L(2 l∞,θ2) consisting of symmetric bilinear forms. Let F = L(2 l∞,θ2) or Ls(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<inf>Ls</inf>(2 l∞,θ2) = ext B<inf>L(</inf>2 l∞,θ2) ∩L<inf>s</inf> (2 l∞,θ2) and expBL<inf>s</inf>(2 l∞,θ2) = exp B<inf>L(2</inf> l∞,θ2) ∩ L<inf>s</inf> (2 l∞,θ2). We classify the smooth points of the closed unit ball of F. It is shown that sm B<inf>L(2</inf> l∞,θ2) ∩L<inf>s</inf> (2 l∞,θ2)⊊ smBL<inf>s</inf>(2 l∞,θ2). As corol-lary we extend the results of [18, 35]. © 2022, Transilvania University of Brasov 1. All rights reserved.

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