Supremum norms for quadratic polynomials

摘要

We define two norms on ${Bbb R}^3$ as follows. For $(a,b,c) in {Bbb R}^3,$ we let $Vert (a,b,c)Vert _{Bbb R}equiv sup , { |ax^2 + bx + c|, :, x in [-1,1]} $ and $Vert (a,b,c)Vert _{Bbb C}equiv sup , { |az^2 + bz + c|:$ $, zin {Bbb C}, |z| leq 1} .$ Geometric properties of these norms are investigated. In particular, we give explicit formulas for these norms, describe the extreme points of the corresponding unit balls, give a parametric description of the unit spheres, and plot their images.