Extreme homogeneous trinomials on the unit square

Abstract

In this paper we investigate the unit ball of the norm in R3 defined as‖(a,b,c)‖□,m,n=sup⁡{|axm+bxm−nyn+cym|:(x,y)∈[0,1]2} for m,n∈N with m>n. In particular we provide a formula to calculate the norm, a parametrization of the unit sphere and an explicit description of the extreme points of the unit ball. Our results extend the study on the unit ball of the space of quadratic forms on □=[0,1]2, namely P(□2), published in [13].Using an elementary change of variables our results also characterize the geometry of the unit ball of the norm in R3 given by|||(a,b,c)|||m,n=sup⁡{|axm+bxm−nyn+cym|:(x,y)∈[−1,1]2}, where m and n are even natural numbers. The norm |||⋅|||m,n has been studied in [23], but only for the case where m is odd.