Supremum Norms for 2-Homogeneous Polynomials on Circle Sectors

We consider the Banach space of two homogeneous polynomials endowed with the supremum norm $\|\cdot\|_{D(\beta)}$ over circle sectors $D(\beta)$ of angle $\beta$ for several values of $\beta\in[0,2\pi]$. We provide an explicit formula for $\|\cdot\|_{D(\beta)}$, a full description of the extreme points of the corresponding unit balls, and a parametrization and a plot of their unit spheres. This work is an extension of a series of papers on the same topic published in the last decade and it has a number of applications to obtain polynomial-type inequalities.