Notions of Convexity

The purpose of this book is to discuss notions of convexity that are assumed beyond calculus and linear algebra. The major part of the book should be accessible to a graduate student as an introduction to complex analysis; the last sections are mainly for those familiar with microlocal analysis.

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The first two chapters of this book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions, pseudoconvex sets, and sets which are convex for supports or singular supports with respect to a differential operator. In addition, the convexity conditions which are relevant for local or global existence of holomorphic differential equations are discussed.