Preface

Summary

Although convexity is used in many different branches of mathematics there is no easily available account dealing with the subject in a manner which combines generality with simplicity. My aim in writing this tract has been to provide a short introduction to this field of knowledge for the use of those starting research or for those working on other topics who feel the need to use and understand convexity.

In a short tract, on a subject such as this, it is difficult to decide both the level of generality to aim at and the exact parts of the subject to omit. On the one hand, to accommodate the needs of economists and others, it is desirable to have available results that refer to n-dimensional real Euclidean space; on the other hand, more general spaces present such diverse characteristics that they cannot be conveniently dealt with in a tract of this size. For this reason the containing space is taken to be n-dimensional real Euclidean space except in the last two chapters. As to the subjects omitted there is nothing on the geometry of numbers, packing or covering problems, differential geometry on convex surfaces, integral geometry or the analogy with complex convexity.

The tract falls naturally into three parts. The first and third chapters contain the basic properties of individual convex sets and functions. The second chapter is an illustration of the way in which the comparatively simple properties obtained in the first chapter can be applied.