Geometry of homogeneous polynomials in ${\mathbb R}^2$

TL;DR

This paper thoroughly investigates the geometry of unit spheres in Banach spaces of homogeneous polynomials in two variables, providing explicit descriptions, extreme points, and solving open questions in the field.

Contribution

It offers a complete geometric characterization of these polynomial spaces, including explicit formulas and techniques, advancing understanding of their structure.

Findings

Complete description of unit spheres in the spaces

Identification of extreme points of the unit balls

Explicit formulas for polynomial norms

Abstract

This work is a thorough and detailed study on the geometry of the unit sphere of certain Banach spaces of homogeneous polynomials in ${\mathbb{R}}^2$. Specifically, we provide a complete description of the unit spheres, identify the extreme points of the unit balls, derive explicit formulas for the corresponding polynomial norms, and describe the techniques required to tackle these questions. To enhance the comprehensiveness of this work, we complement the results and their proofs with suitable diagrams and figures. The new results presented here settle some open questions posed in the past. For the sake of completeness of this work, we briefly discuss previous known results and provide directions of research and applications of our results.