Geometry of Banach spaces of trinomials

For each pair of numbers m,n ∈ N with m>n, we consider the norm on R3 given by (a, b, c) m,n = sup{|axm + bxn + c|: x ∈ [−1, 1]} for every (a, b, c) ∈ R3. We investigate some geometrical properties of these norms. We provide an explicit formula for · m,n, a full description of the extreme points of the corresponding unit balls and a parametrization and a plot of their unit spheres. © 2007 Elsevier Inc. All rights reserved.