# Fundamentals of Convex Analysis

Springer Science & Business Media, Dec 6, 2012 - Mathematics - 259 pages
This book is an abridged version of our two-volume opus Convex Analysis and Minimization Algorithms [18], about which we have received very positive feedback from users, readers, lecturers ever since it was published - by Springer-Verlag in 1993. Its pedagogical qualities were particularly appreciated, in the combination with a rather advanced technical material. Now [18] hasa dual but clearly defined nature: - an introduction to the basic concepts in convex analysis, - a study of convex minimization problems (with an emphasis on numerical al- rithms), and insists on their mutual interpenetration. It is our feeling that the above basic introduction is much needed in the scientific community. This is the motivation for the present edition, our intention being to create a tool useful to teach convex anal ysis. We have thus extracted from [18] its "backbone" devoted to convex analysis, namely ChapsIII-VI and X. Apart from some local improvements, the present text is mostly a copy of the corresponding chapters. The main difference is that we have deleted material deemed too advanced for an introduction, or too closely attached to numerical algorithms. Further, we have included exercises, whose degree of difficulty is suggested by 0, I or 2 stars *. Finally, the index has been considerably enriched. Just as in [18], each chapter is presented as a "lesson", in the sense of our old masters, treating of a given subject in its entirety.

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### Contents

 Notation Elementary Results 2 Convex Sets Attached to a Convex Set 33 Projection onto Closed Convex Sets 46 Conical Approximations of Convex Sets 62 Exercises 70 Functional Operations Preserving Convexity 87 Local and Global Behaviour of a Convex Function 102 First and SecondOrder Differentiation 1 10 110
 Sublinear Functions 123 The Support Function of a Nonempty Set 134 Correspondence Between Convex Sets and Sublinear Functions 143 Exercises 161 Subdifferentials of Finite Convex Functions 164 E Conjugacy in Convex Analysis 210 Bibliographical Comments 245 Copyright

 Exercises 117