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Notions of convexity. (English) Zbl 0835.32001
Progress in Mathematics (Boston, Mass.). 127. Basel: Birkhäuser. viii, 414 p. (1994).
The goal of this book is to show how various notions and results from analysis follow from the general notion of convexity. Most of the results presented are from the fields of complex analysis (one and several variables) and partial differential equations, but there are also a few (some inevitable) examples from geometry, game theory, and special functions.
Chapters I and II present the classical theory of convex functions of one and several variables respectively. The latter includes a section on convexity in Fourier analysis, and there are sections on less standard topics like projective convexity, semi-convexity and quasi-convexity.
The presentation of Chapter III on subharmonic functions parallels that of Chapter I with harmonic functions replacing linear functions. The basic tool used in the applications is the Riesz representation theorem. Several convexity results, in the spirit of the Hadamard three circle theorem, are proved, with classical applications to \(H^p\) spaces, but also more unusual results like a convexity proof of the Koebe 1/4- theorem.
Chapter IV covers the basic theory of plurisubharmonic functions and pseudo-convex domains. The material covered is pretty much that of Chapter 4 of the author’s ‘An introduction to complex analysis in several variables’. 3rd rev. ed., North Holland, Amsterdam (1990; Zbl 0685.32001).
Chapter V is somehow more experimental. After observing that the classes of convex, subharmonic, and plurisubharmonic functions are invariant under composition with functions from the full linear group, the orthogonal group, and the full complex linear group respectively, the author asks what happens if other subgroups are considered. This is more tentative, and no applications are given.
The question of finding conditions on an open set \(X\) that will ensure that a linear partial differential equation with constant coefficients \(P(D)u = f\) has a solution for any \(f\) in a given class leads to the notion of \(P\)-convexity for supports and for singular supports. The main existence theorem is proved in the author’s ‘The analysis of linear partial differential operators’, part I–IV, Springer Verlag, Berlin (1983-85; Zbl 0521.35001; Zbl 0521.35008; Zbl 0601.35001; Zbl 0612.35001).
In Chapter VI of the book under review, the emphasis is shifted to the geometric characterisation of \(P\)-convexity. It is precisely in convex domains that all such PDEs are solvable. The cases of operators of the form \(P (\partial/ \partial \overline z_1, \dots, \partial/ \partial \overline z_n)\) and of analytic differential equations, where pseudo- convexity guarantees \(P\)-convexity, are also studied, using the techniques of Chapter IV.
Chapter VII presents recent results on the local solvability by analytic functions of the equation \(\partial u/ \partial z_1 = f\) in strictly pseudoconvex domains near boundary points. It requires some knowledge of microlocal analysis.
This book will provide the mature graduate student with an excellent reference for subharmonic functions and the theory of functions of several complex variables. It will also be of interest to the researcher in complex analysis or partial differential equations for the new material it contains that has not appeared in book form before. This includes section 1.7 on conditions for the supremum of a one-parameter family of convex functions to be convex, its analogue results for subharmonic functions, Chapter V, and Chapter VII, which is based on the thesis of J.-M. Trépreau.
I think that the author has succeeded in showing the universality of the notion of convexity and this book should be inspiring for any analyst. If something negative had to be said, one could deplore the fact that a 400- page book containing such a wealth of results has been provided with a pathetic 2-page index.

32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32U05 Plurisubharmonic functions and generalizations
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31-02 Research exposition (monographs, survey articles) pertaining to potential theory
32T99 Pseudoconvex domains
31C10 Pluriharmonic and plurisubharmonic functions
26B25 Convexity of real functions of several variables, generalizations
26A51 Convexity of real functions in one variable, generalizations
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators