Geometry of spaces of homogeneous trinomials on $${\\mathbb {R}}^2$$

Abstract

For each pair of numbers $$m,n\\in {{\\mathbb {N}}}$$\n with $$m>n$$\n , we consider the norm on $${{\\mathbb {R}}}^3$$\n given by $$\\Vert (a,b,c)\\Vert _{m,n}=\\sup \\{|ax^m+bx^{m-n}y^n+cy^m|:x,y\\in [-1,1]\\}$$\n for every $$(a,b,c)\\in {{\\mathbb {R}}}^3$$\n . We investigate some geometrical properties of these norms. We provide an explicit formula for $$\\Vert \\cdot \\Vert _{m,n}$$\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.