Convexity in complex analysis

Abstract

Convexity is the key concept of functional analysis, but apart from some notable exceptions, it has played a relatively minor role in several complex variables theory. The work of Lempert has focused attention on convex domains, and in these two lectures I will present examples involving invariant metrics in complex analysis where convexity, whether realized as in the case of bounded symmetric domains or assumed as in the case of Blp , is essential. Functional analysis brings to problems a variety of developed concepts such as complex extreme points, complex uniform convexity and an approach which is often coordinate and dimension free. I hope to illustrate these points in my lectures. Throughout this article X will denote a Banach space over the complex numbers C, BX will denote the open unit ball in X and BX its closure. We denote by ∆ the open unit disc in C. Our first example relates the maximum modulus theorem of complex analysis with the functional analytic concept of complex extreme point. A point x in X, (‖x‖ = 1), is a complex extreme point (of the unit ball) if ‖x + λy‖ ≤ 1 for all λ ∈ ∆ implies y = 0. The strong maximum modulus theorem, due to Thorp and Whitely in 1965, states the following: