Notions of Convexity

by Lars Hörmander

The term convexity used to describe these lectures given at the Univer sity of Lund in 1991-92 should be understood in a wide sense. Only Chap ters I and II are devoted to convex sets and functions in the traditional sense of convexity. The following chapters study other kinds of convexity which occur in analysis. Most prominent is the pseudo-convexity (plurisubh- monicity) in the theory of functions of several complex variables discussed in Chapter IV. It relies on the theory of subharmonic functions in R DEGREES, so Chapter III is devoted to subharmonic functions in R" DEGREES for any n. Existence theorems for constant coefficient partial differential operators in R' DEGREES are re lated to various kinds of convexity conditions, depending on the operator. Chapter VI gives a survey of the rather incomplete results which are known on their geometrical meaning. There are also natural classes of "convex" functions related to subgroups of the linear group, which specialize to sev eral of the notions already mentioned. They are discussed in Chapter V. The last chapter. Chapter VII, is devoted to the conditions for solvability of microdifferential equations, which can also be considered as a branch of convexity theory. The whole chapter is an exposition of a part of the thesis of J.-

416 pages, Hardcover

First published November 1, 1994

About the author

Lars Valter Hörmander (born 24 January 1931) is a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". He was awarded the Fields Medal in 1962, the Wolf Prize in 1988, and the Leroy P. Steele Prize in 2006. His Analysis of Linear Partial Differential Operators I–IV is considered a standard work on the subject of linear partial differential operators.

Hörmander completed his Ph.D. in 1955 at Lund University. Hörmander then worked at Stockholm University, at Stanford University, and at the Institute for Advanced Study in Princeton, New Jersey. He returned to Lund University as professor from 1968 until 1996, when he retired with the title of professor emeritus.

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