UNIVERSITY OF KAISERSLAUTERN

Department of Mathematics

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@f (x) IRn \Theta  f\Gamma 1g

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Convex Analysis

Dr. Stefan Nickel

Preface Convex analysis is one of the mathematical tools which is used both explicitly and indirectly in many mathematical disciplines. However, there are not so many courses which have convex analysis as the main topic. More often, parts of convex analysis are taught in courses like linear or nonlinear optimization, probability theory, geometry, location theory, etc.. This manuscript gives a systematic introduction to the concepts of convex analysis. A focus is set to the geometrical interpretation of convex analysis. This focus was one of the reasons why I have decided to restrict myself to the finite dimensional case. Another reason for this restriction is that in the infinite dimensional case many proofs become more difficult and more technical. Therefore, it would not have been possible (for me) to cover all the topics I wanted to discuss in this introductory text in the infinite dimensional case, too. Anyway, I am convinced that even for someone who is interested in the infinite dimensional case this manuscript will be a good starting point. When I offered a course in convex analysis in the Wintersemester 1997/1998 (upon which this manuscript is based) a lot of students asked me how this course fits in their own studies. Because this manuscript will (hopefully) be used by some students in the future, I will give here some of the possible statements to answer this very question.

ffl Convex analysis can be seen as an extension of classical analysis, in

which still we get many of the results, like a mean-value theorem, with less assumptions on the smoothness of the function.

ffl Convex analysis can be seen as a foundation of linear and nonlinear

optimization which provides many tools to handle concepts in optimization much easier (for example the Lemma of Farkas).

ffl Finally, convex analysis can be seen as a link between abstract geometry

and very algorithmic oriented computational geometry.

As already explained before, this manuscript is based on a one semester course and therefore cannot cover all topics and discuss all aspects of convex analysis in detail. To guide the interested reader I have included a list of nice books about this subject at the end of the manuscript. It should be noted that the philosophy of this course follows [3], [4] and THE BOOK of modern convex analysis [6]. The geometrical emphasis however, is also related to intentions of [1].

iii At the end of this preface I would like to thank Dr. Matthias Ehrgott for preparing the exercises for this manuscript and his general support. I also would like to thank Chokri Hamdaoui, who typed this manuscript with a great enthusiasm. Last but not least I thank all the members of the research group of Prof. Hamacher for their help in proof-reading and for their useful remarks.

Stefan Nickel, October 1998

iv Contents I CONVEX SETS 1 1 BASIC CONCEPTS 3

1.1 Definition and Important Examples . . . . . . . . . . . . . . . 3 1.2 Operations on Convex Sets Preserving Convexity . . . . . . . 7 1.3 Convex Combinations and Convex Hulls . . . . . . . . . . . . 11 1.4 Closed Convex Sets and Hulls . . . . . . . . . . . . . . . . . . 20 1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23