Supremum norms for quadratic polynomials (Q5929496)

The following two norms on \(\mathbb{R}^3\) are investigated: \[ \bigl \|(a,b,c) \bigr\|_\mathbb{R}= \sup\bigl\{|ax^2+ bx+c|:x\in[-1,1] \bigr\}, \] \[ \bigl\|(a,b,c)\bigr\|_\mathbb{C} =\sup\bigl\{|ax^2+bx+c |:x\in\mathbb{C}, |x|\leq 1\bigr\}. \] The explicit formulae for the norms are given, the extreme points of the unit balls are described, the parametric descriptions of the unit spheres are obtained.