Notions of Convexity
This is a reprint of the 1994 edition of the same book devoted to convexity and its applications to analysis. Convexity is meant in rather an abstract sense: a function is convex if it attains its maximum always on the boundary of a given domain, even if another function from some linear space L (of functions satisfying some specific properties, like linearity) is subtracted from it. For L being the space of affine functions, such a notion is equivalent to the usual convexity and the first two chapters of the book develop the classical theory of convex functions and sets. With other choices of L, deep analogies and generalisations of ordinary convexity are obtained. Thus, chapter 3 develops the theory of subharmonic functions (L being, in this case, the linear space of all harmonic functions). Based on this, chapter 4 presents the theory of plurisubharmonic functions and related notions like pseudoconvexity. These concepts have many important applications in partial differential equations, the theory of functions of several complex variables and harmonic analysis.
Chapters 5, 6 and 7 are devoted to more special topics (some of them being systematically developed in a book for the first time) like convexity with respect to a linear group and convexity with respect to differential operators. The book concludes with an exposition of the Trépreau results on microlocal analysis. Undoubtedly, this monograph written by a master of the field will be useful not only for specialists in analysis but also (in particular the first chapters; the rest of the book requires familiarity with distributions, differential geometry and pseudodifferential calculus) to students willing to learn this important, classical but still lively subject where so many prominent mathematicians of the last century made their contribution.