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.