cartea buna

8/3/2019 cartea buna

http://slidepdf.com/reader/full/cartea-buna 1/14

Canadian Mathematical Society

Société mathématique du Canada

Editors-in-chief Rédacteurs-en-chef J. Borwein

K.Dilcher

Advisory Board Comité consultatif

P. Borwein

R.Kane

S.Shen



CMS Books in MathematicsOuvrages de mathématiques de la SMC

1 HERMAN /KUCERA / SIMSA Equations and Inequalities

2 ARNOLD Abelian Groups and Representations of Finite Partially Ordered Sets

3 BORWEIN /LEWIS Convex Analysis and Nonlinear Optimization

4 LEVIN /LUBINSKY Orthogonal Polynomials for Exponential Weights

5 KANE Reflection Groups and Invariant Theory

6 PHILLIPS Two Millennia of Mathematics

7 DEUTSCH Best Approximation in Inner Product Spaces8 FABIAN ET AL. Functional Analysis and Infinite-Dimensional Geometry

9 KRÍZEK /LUCA /SOMER 17 Lectures on Fermat Numbers

10 BORWEIN Computational Excursions in Analysis and Number Theory

11 REED /SALES (Editors) Recent Advances in Algorithms and Combinatorics

12 HERMAN /KUCERA / SIMSA Counting and Configurations

13 NAZARETH Differentiable Optimization and Equation Solving14 PHILLIPS Interpolation and Approximation by Polynomials

15 BEN-ISRAEL /GREVILLE Generalized Inverses, Second Edition

16 ZHAO Dynamical Systems in Population Biology

17 GÖPFERT ET AL. Variational Methods in Partially Ordered Spaces

18 AKIVIS /GOLDBERG Differential Geometry of Varieties with DegenerateGauss Maps

19 MIKHALEV /SHPILRAIN /YU Combinatorial Methods

20 BORWEIN /ZHU Techniques of Variational Analysis

21 VAN BRUMMELEN /KINYON Mathematics and the Historian’s Craft: TheKenneth O. May Lectures

22 LUCCHETTI Convexity and Well-Posed Problems

23 NICULESCU /PERSSON Convex Functions and Their Applications

24 SINGER Duality for Nonconvex Approximation and Optimization25 HIGGINSON /PIMM /SINCLAIR Mathematics and the Aesthetic



Constantin P. NiculescuLars-Erik Persson

Convex Functions andTheir Applications

A Contemporary Approach

With 8 Illustrations



Constantin P. Niculescu Lars-Erik PerssonDepartment of Mathematics Department of MathematicsUniversity of Craiova Luleå University of TechnologyCraiova 200585 Luleå 97187

Romania [email protected] [email protected]

Editors-in-Chief Rédacteurs-en-chef Jonathan BorweinKarl DilcherDepartment of Mathematics and StatisticsDalhousie UniversityHalifax, Nova Scotia B3H 3J5

[email protected]

Mathematics Subject Classification (2000): 26Axx, 26Bxx, 26Dxx, 46Axx, 52Axx

Library of Congress Control Number: 2005932044

ISBN-10: 0-387-24300-3ISBN-13: 978-0387-24300-9

Printed on acid-free paper.© 2006 Springer Science+Business Media, Inc.All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if they arenot identified as such, is not to be taken as an expression of opinion as to whether or not they are subjectto proprietary rights.

Printed in the United States of America. (MVY)

9 8 7 6 5 4 3 2 1

springeronline.com



To Liliana and Lena



Preface

It seems to me that the notion of convex function is

just as fundamental as positive function or increasing

function. If I am not mistaken in this, the notion ought

to find its place in elementary expositions of the theory

of real functions.

J. L. W. V. Jensen

Convexity is a simple and natural notion which can be traced back toArchimedes (circa 250 B.C.), in connection with his famous estimate of thevalue of π (by using inscribed and circumscribed regular polygons). He no-ticed the important fact that the perimeter of a convex figure is smaller thanthe perimeter of any other convex figure surrounding it.

As a matter of fact, we experience convexity all the time and in manyways. The most prosaic example is our upright position, which is secured as

long as the vertical projection of our center of gravity lies inside the convexenvelope of our feet. Also, convexity has a great impact on our everyday lifethrough numerous applications in industry, business, medicine, and art. Sodo the problems of optimum allocation of resources and equilibrium of non-cooperative games.

The theory of convex functions is part of the general subject of convexity,since a convex function is one whose epigraph is a convex set. Nonethelessit is an important theory per se, which touches almost all branches of math-ematics. Graphical analysis is one of the first topics in mathematics which

requires the concept of convexity. Calculus gives us a powerful tool in recog-nizing convexity, the second-derivative test. Miraculously, this has a naturalgeneralization for the several variables case, the Hessian test. Motivated bysome deep problems in optimization and control theory, convex function the-ory has been extended to the framework of infinite dimensional Banach spaces(and even further).

The recognition of the subject of convex functions as one that deservesto be studied in its own right is generally ascribed to J. L. W. V. Jensen[114], [115]. However he was not the first to deal with such functions. Amonghis predecessors we should recall here Ch. Hermite [102], O. Holder [106] andO. Stolz [233]. During the twentieth century, there was intense research ac-tivity and significant results were obtained in geometric functional analysis,mathematical economics, convex analysis, and nonlinear optimization. A clas-



VIII Preface

sic book by G. H. Hardy, J. E. Littlewood and G. P olya [99] played a largerole in the popularization of the subject of convex functions.

Roughly speaking, there are two basic properties of convex functions thatmake them so widely used in theoretical and applied mathematics:

• The maximum is attained at a boundary point.

• Any local minimum is a global one. Moreover, a strictly convex function

admits at most one minimum.

The modern viewpoint on convex functions entails a powerful and elegantinteraction between analysis and geometry. In a memorable paper dedicatedto the Brunn–Minkowski inequality, R. J. Gardner [88, p. 358], described thisreality in beautiful phrases: [convexity] “appears like an octopus, tentaclesreaching far and wide, its shape and color changing as it roams from one area

to the next. It is quite clear that research opportunities abound.”Over the years a number of notable books dedicated to the theory and ap-plications of convex functions appeared. We mention here: L. Hormander [108],M. A. Krasnosel’skii and Ya. B. Rutickii [132], J. E. Pecaric, F. Proschan andY. C. Tong [196], R. R. Phelps [199], [200] and A. W. Roberts and D. E. Var-berg [212]. The references at the end of this book include many other finebooks dedicated to one aspect or another of the theory.

The title of the book by L. Hormander, Notions of Convexity, is verysuggestive for the present state of art. In fact, nowadays the study of convexfunctions has evolved into a larger theory about functions which are adaptedto other geometries of the domain and/or obey other laws of comparison of means. Examples are log-convex functions, multiplicatively convex functions,subharmonic functions, and functions which are convex with respect to asubgroup of the linear group.

Our book aims to be a thorough introduction to contemporary convexfunction theory. It covers a large variety of subjects, from the one real vari-able case to the infinite dimensional case, including Jensen’s inequality and itsramifications, the Hardy–Littlewood–Polya theory of majorization, the the-ory of gamma and beta functions, the Borell–Brascamp–Lieb form of the

Prekopa–Leindler inequality (as well as the connection with isoperimetric in-equalities), Alexandrov’s well-known result on the second differentiability of convex functions, the highlights of Choquet’s theory, a brief account on therecent solution to Horn’s conjecture, and many more. It is certainly a bookwhere inequalities play a central role but in no case a book on inequalities.Many results are new, and the whole book reflects our own experiences, bothin teaching and research.

This book may serve many purposes, ranging from a one-semester gradu-ate course on Convex Functions and Applications to additional bibliographic

material. In a course for first year graduate students, we used the followingroute:

• Background : Sections 1.1–1.3, 1.5, 1.7, 1.8, 1.10.• The beta and gamma functions : Section 2.2.



Preface IX

• Convex functions of several variables: Sections 3.1–3.12.• The variational approach of partial differential equations : Appendix C.

The necessary background is advanced calculus and linear algebra. This

can be covered from many sources, for example, from Analysis I and II byS. Lang [137], [138]. A thorough presentation of the fundamentals of measuretheory is also available in L. C. Evans and R. F. Gariepy [74]. For furtherreading we recommend the classical texts by F. H. Clarke [56] and I. Ekelandand R. Temam [70].

Our book is not meant to be read from cover to cover. For example, Sec-tion 1.9, which deals with the Hermite–Hadamard inequality, offers a goodstarting point for Choquet’s theory. Then the reader may continue with Chap-ter 4, where this theory is presented in a slightly more general form, to allow

the presence of certain signed measures. We recommend this chapter to bestudied in parallel with the Lectures on Choquet’s theory by R. R. Phelps[200]. For the reader’s convenience, we collected in Appendix A all the nec-essary material on the separation of convex sets in locally convex Hausdorff spaces (as well as a proof of the Krein–Milman theorem).

Appendix B may be seen both as an illustration of convex function theoryand an introduction to an important topic in real algebraic geometry: thetheory of semi-algebraic sets.

Sections 3.11 and 3.12 offer all necessary background on a further study of convex geometric analysis, a fast-growing topic which relates many importantbranches of mathematics.

To help the reader in understanding the theory presented, each sectionends with exercises (accompanied by hints). Also, each chapter ends withcomments covering supplementary material and historical information. Theprimary sources we have relied upon for this book are listed in the references.

In order to avoid any confusion relative to our notation, a symbol indexwas added for the convenience of the reader. Notice that our book deals onlywith real linear spaces and all Borel measures under attention are assumed tobe regular .

We wish to thank all our colleagues and friends who read and commentedon various versions and parts of the manuscript: Madalina Deaconu, AndaluziaMatei, Sorin Micu, Florin Popovici, Mircea Preda, Thomas Stromberg, AndreiVernescu, Peter Wall, Anna Wedestig and Tudor Zamfirescu.

We also acknowledge the financial support of Wenner–Gren Foundations(Grant 25 12 2002), which made possible the cooperation of the two authors.

In order to keep in touch with our readers, a web page for this book willbe made available at http://www.inf.ucv.ro/∼niculescu/Convex Functions.html

Craiova and Lulea Constantin P. Niculescu

September 2004 Lars-Erik Persson



Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII

List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XIII

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Convex Functions on Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1 Convex Functions at First Glance . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Young’s Inequality and Its Consequences . . . . . . . . . . . . . . . . . . . 141.3 Smoothness Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4 An Upper Estimate of Jensen’s Inequality . . . . . . . . . . . . . . . . . . 271.5 The Subdifferential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.6 Integral Representation of Convex Functions . . . . . . . . . . . . . . . . 361.7 Conjugate Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.8 The Integral Form of Jensen’s Inequality . . . . . . . . . . . . . . . . . . . 441.9 The Hermite–Hadamard Inequality . . . . . . . . . . . . . . . . . . . . . . . . 501.10 Convexity and Majorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531.11 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2 Comparative Convexity on Intervals . . . . . . . . . . . . . . . . . . . . . . . 652.1 Algebraic Versions of Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.2 The Gamma and Beta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 682.3 Generalities on Multiplicatively Convex Functions . . . . . . . . . . . 772.4 Multiplicative Convexity of Special Functions . . . . . . . . . . . . . . . 832.5 An Estimate of the AM–GM Inequality . . . . . . . . . . . . . . . . . . . . 852.6 (M, N )-Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882.7 Relative Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

2.8 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97



XII Contents

3 Convex Functions on a Normed Linear Space . . . . . . . . . . . . . . 1013.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.2 The Orthogonal Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.3 Hyperplanes and Separation Theorems . . . . . . . . . . . . . . . . . . . . . 1093.4 Convex Functions in Higher Dimensions . . . . . . . . . . . . . . . . . . . . 1123.5 Continuity of Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193.6 Positively Homogeneous Functions . . . . . . . . . . . . . . . . . . . . . . . . . 1233.7 The Subdifferential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283.8 Differentiability of Convex Functions . . . . . . . . . . . . . . . . . . . . . . . 1353.9 Recognizing Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413.10 The Convex Programming Problem . . . . . . . . . . . . . . . . . . . . . . . . 1453.11 Fine Properties of Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . 1523.12 Prekopa–Leindler Type Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 158

3.13 Mazur–Ulam Spaces and Convexity . . . . . . . . . . . . . . . . . . . . . . . . 1653.14 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

4 Choquet’s Theory and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1774.1 Steffensen–Popoviciu Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1774.2 The Jensen–Steffensen Inequality and Majorization . . . . . . . . . . 1844.3 Steffensen’s Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1904.4 Choquet’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1924.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

A Background on Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203A.1 The Hahn–Banach Extension Theorem . . . . . . . . . . . . . . . . . . . . . 203A.2 Separation of Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207A.3 The Krein–Milman Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

B Elementary Symmetric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 213B.1 Newton’s Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213B.2 More Newton Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217B.3 A Result of H. F. Bohnenblust . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

C The Variational Approach of PDE . . . . . . . . . . . . . . . . . . . . . . . . . 223C.1 The Minimum of Convex Functionals . . . . . . . . . . . . . . . . . . . . . . 223C.2 Preliminaries on Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 226C.3 Applications to Elliptic Boundary-Value Problems . . . . . . . . . . . 228C.4 The Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

D Horn’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233D.1 Weyl’s Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

D.2 The Case n = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237D.3 Majorization Inequalities and the Case n = 3 . . . . . . . . . . . . . . . 238

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253



List of symbols

N, Z, Q, R, C: the classical numerical sets (naturals, integers etc.)

N: the set of positive integers

R+: the set of nonnegative real numbers

R+: the set of positive real numbers

R: the set of extended real numbers

∅: empty set

∂A: boundary of A

A: closure of A

intA: interior of A

A: polar of A

rbd(A): relative boundary of A

ri(A): relative interior of A

Br(a): open ball center a, radius r

Br(a): closed ball center a, radius r

[x, y]: line segment

aff(A): affine hull of A

co(A): convex hull of A

co(A): closed convex hull of A

ExtK : the extreme points of K

λA + µB = λx + µy | x ∈ A, y ∈ B

|A

|: cardinality of A

diam(A): diameter of A

Voln(K ): n-dimensional volume



XIV List of symbols

P C (x): set of best approximation from x

χA: characteristic function of A

id: identityδC : indicator function

f |K : restriction of f to K

f ∗: (Legendre) conjugate function

f : lower envelope of f

f : upper envelope of f

f ↓: symmetric-decreasing rearrangement of f

dU (x) = d(x,U ): distance from x to U

P K(x): orthogonal projection

dom(f ): effective domain of f

epi(f ): epigraph of f

graph(f ): graph of f

∂f (a): subdifferential of f at a

supp(f ): support of f

f ∗ g: convolution

f g: infimal convolution

Rn: Euclidean n-space

Rn+ = (x1, . . . , xn) ∈ Rn | x1, . . . , xn ≥ 0, the nonnegative orthant

Rn++ = (x1, . . . , xn) ∈ Rn | x1, . . . , xn > 0

Rn≥ = (x1, . . . , xn) ∈ Rn | x1 ≥ · · · ≥ xn

x, y, x · y: inner product

Mn(R), Mn(C): spaces of n × n-dimensional matrices

GL(n,R): the group of nonsingular matrices

Sym+(n,R): the set of all positive matrices of Mn(R)

Sym++(n,R): the set of all strictly positive matrices of Mn(R)

dim E : dimension of E

E : dual space

A: adjoint matrix

det A: determinant of A

kerA: kernel (null space) of Arng A: range of A

traceA: trace of A



List of symbols XV

lim inf x→a

f (x) = limr→0 inf f (x) | x ∈ dom(f ) ∩ Br(a): lower limit

lim supx→a

f (x) = limr→0 supf (x) | x ∈ dom(f ) ∩ Br(a): upper limit

Df (a) and Df (a): lower and upper derivativesD2f (a) and D

2f (a): lower and upper second symmetric derivatives

f +(a; v) and f −(a; v): lateral directional derivatives

f (a; v): first Gateaux differential

f (a; v, w): second Gateaux differential

df : first order differential

d2f : second order differential

∂f

∂xk : partial derivative

Dαf =∂ α1+···+αnf

∂xα11 · · · ∂xαnn∇: gradient

HessA f : Hessian matrix of f at a

∇2f (a): Alexandrov Hessian of f at a

A(K ): space of real-valued continuous and affine functions

Conv(K ): space of real-valued continuous and convex functions

C (K ): space of real-valued continuous functions

C m(Ω) = f | Dαf ∈ C (Ω) for all |α| ≤ m

C m(Ω) = f | Dαf uniformly continuous on Ω for all |α| ≤ m

C ∞c (Ω): space of functions of class C ∞ with compact support

L p(Ω): space of p-th-power Lebesgue integrable functions on Ω

L p(µ): space of p-th-power µ-integrable functions

f Lp

: L p

-normess sup: essential supremum

p: space of p-th-power absolutely summing real sequences

Lip(f ): Lipschitz constant

H m(Ω): Sobolev space on Ω

f H m : Sobolev norm

H m0 (Ω): norm closure of C ∞c (Ω) in H m(Ω)

Prob(X ): set of Borel probability measures on X

δa: Dirac measure concentrated at a

xµ: barycenter of µ



XVI List of symbols

E (f ) : conditional expectation of f

A(s, t), G(s, t), H (s, t): arithmetic, geometric and harmonic means

I (s, t): identric meanL(s, t): logarithmic mean

M p(s, t), M p(f ;µ): Holder (power) mean

M [ϕ]: quasi-arithmetic mean

cartea buna

Documents

Transcript of cartea buna