Geometry of Spaces of Polynomials

Connections between the shape of the unit ball of a Banach space and analytic properties of the Banach space have been studied for many years. In this article, some geometric properties of spaces related to n-homogeneous polynomials are considered. In particular, the rotundity and smoothness of spaces of continuous n-homogeneous polynomials and its preduals are studied. Furthermore, an inequality relating the product of the norms of linear functionals on a Banach space with the norm of the continuous n-homogeneous polynomial determined by the product of the linear functionals is derived. This inequality is used to study the strongly exposed points of the predual of the space of continuous 2-homogeneous polynomials.