Convex analysis is that special branch of mathematics which directly borders
onto classical (smooth) analysis on the one side and geometry on the other.
Almost all mathematicians (and very many practitioners) must have the skills to
work with convex sets and functions, and extremal problems, since convexity
continually crops up in the investigation of very diverse problems in mathematics
and the natural sciences. It seems that some of the elements of convex analysis
must occupy a place in mathematical education at any level. Bearing all of this
in mind, this article was written to take account of a very broad circle of readers.
In particular, this is true of the introduction, where an attempt has been made
to explain the fundamental concepts, ideas, statements of problems and the
essence of the matter to any interested person.
Chapters 1-3 are addressed to those who wish to become acquainted with the
foundations of convex analysis and those areas of mathematics where convex
analysis is applied. Chapter 4 is devoted to a brief survey of the areas bordering
convex analysis which have been intensively developed in recent times.
The author has not set himself the aim of encompassing the whole of convex
analysis; his endeavour has been to discuss the sources, the fundamental concepts
and results, to relate the most important applications of convexity and to
mention some of the most significant and beautiful theorems.
In conclusion the author his thanks to V.I. Blagodatskikh, A.D. Ioffe,