Geometry of the Unit Sphere in Polynomial Spaces

This brief presents a global perspective on the geometry of spaces of polynomials. Its particular focus is on polynomial spaces of dimension 3, providing, in that case, a graphical representation of the unit ball. Also, the extreme points in the unit ball of several polynomial spaces are characterized. Finally, a number of applications to obtain sharp classical polynomial inequalities are presented. The study performed is the first ever complete account on the geometry of the unit ball of polyno