Geometry of spaces of homogeneous trinomials on ℝ2

Abstract

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{|ax^m + bx^(m−n)y^n + cy^m| ∶ x, y ∈ [−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 for certain values of m and n.