Supremum norms for 2-homogeneous polynomials on circle sectors (Q2922473)

The authors investigate the geometry of the \(2\)-homogeneous polynomial \(P(x,y)= ax^2+by^2+cxy\) on \({\mathbb R}^2\), where the norm of \(P\), \(\|P\|_{D(\beta)}\), is defined as the supremum over the sector \(D(\beta):=\{re^{i\theta}:0\leq r\leq 1\), \(0\leq \theta\leq\beta\}\). Explicit formulae for the unit sphere of these spaces and the extreme points of the unit ball are given when \(\beta\geq \pi\). Although the case \(\beta<\pi\) is more difficult to work with, the authors obtain formulae for unit spheres and the extreme points of the corresponding unit balls when \(\beta={{\pi}\over{4}},{{\pi}\over{2}}\) and \({3\pi}\over{4}\).