Polynomials and entire functions: zeros and geometry of the unit ball

Abstract

We characterize the extreme and exposed points of the unit ball in certain L 1 -spaces of polynomials and entire functions. As usual, an element x ∈ b (X) is said to be an extreme point of b(X )i f it is not a proper convex combination of two distinct points in b (X). Further, an x in b(X) is called an exposed point of b(X) if there exists a functional Φ ∈ X ∗ such that � Φ� = 1 and the set {y ∈ b(X ):Φ (y )=1 } consists of precisely one element, x. Of course, every exposed point is extreme, and every extreme point is of norm one. In this paper, we determine the extreme and exposed points of the unit ball in the following spaces: • the space Pn(T) of all (analytic) polynomials p of degree ≤ n, living on the circle T := {z ∈ C : |z| =1 } and endowed with the L 1 -norm � p� 1,T := 1