## Fundamentals of Convex AnalysisThis 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

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 |

Exercises | 117 |

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### Common terms and phrases

affine function affine hull affine manifold arbitrary assumption calculus rule closed convex cone closed convex function closed convex hull closed convex set closed half-spaces closed sublinear function closure COIV compact set compute conjugate consider contains Conv Conversely convex analysis convex combination convex cone convex hull Corollary defined denote epi f epif epigraph equivalent Euclidean Example exposed face extreme point f is convex finite function f geometric given gradient half-spaces implies inequality infimal convolution infimum intersection Lemma Let f Lipschitz matrix minimization nonempty closed convex nonnegative norm normal cone notation obtain orthogonal positive homogeneity Proof Proposition prove quadratic R U oo relative interior Remark resp result ridom f satisfying scalar product semi-continuous Show Sn(R space subdifferential subgradient sublevel-set sublinear functions subspace support function supremum tangent cone Theorem vector Vf(x