Table Of ContentConvexFunctions:Constructions,Characterizations
andCounterexamples
Likedifferentiability,convexityisanaturalandpowerfulpropertyoffunctionsthat
playsasignificantroleinmanyareasofmathematics,bothpureandapplied.Itties
togethernotionsfromtopology,algebra,geometryandanalysis,andisanimportant
toolinoptimization,mathematicalprogrammingandgametheory.Thisbook,
whichistheproductofacollaborationofover15years,isuniqueinthatitfocuses
onconvexfunctionsthemselves,ratherthanonconvexanalysis.Theauthors
explorethevariousclassesandtheircharacteristics,treatingconvexfunctionsin
bothEuclideanandBanachspaces.
Theybeginbydemonstrating,largelybywayofexamples,theubiquityof
convexity.Chapter2thenprovidesanextensivefoundationforthestudyofconvex
functionsinEuclidean(finite-dimensional)space,andChapter3reprisesimportant
specialstructuressuchaspolyhedrality,selectiontheorems,eigenvalueoptimization
andsemidefiniteprogramming.Chapters4and5playthesamerolein
(infinite-dimensional)Banachspace.Chapter6discussesanumberofotherbasic
topics,suchasselectiontheorems,setconvergence,integralandtraceclass
functionals,andconvexfunctionsonBanachlattices.
Chapters7and8examineLegendrefunctionsandtheirrelationtothegeometry
ofBanachspaces.Thefinalchapterinvestigatestheapplicationofconvexfunctions
to(maximal)monotoneoperatorsthroughtheuseofarecentlydiscoveredclassof
convexrepresentativefunctionsofwhichtheFitzpatrickfunctionistheprogenitor.
The book can either be read sequentially as a graduate text, or dipped into by
researchersandpractitioners. Eachchaptercontainsavarietyofconcreteexamples
and over 600 exercises are included, ranging in difficulty from early graduate to
researchlevel.
EncyclopediaofMathematicsandItsApplications
Allthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfrom
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73 M.SternSemimodularLattices
74 I.LasieckaandR.TriggianiControlTheoryforPartialDifferentialEquationsI
75 I.LasieckaandR.TriggianiControlTheoryforPartialDifferentialEquationsII
76 A.A.IvanovGeometryofSporadicGroupsI
77 A.SchinzelPolynomialswithSpecialRegardtoReducibility
78 T.Beth,D.JungnickelandH.LenzDesignTheoryII,2ndedn
79 T.W.PalmerBanachAlgebrasandtheGeneralTheoryof*-AlgebrasII
80 O.StormarkLie’sStructuralApproachtoPDESystems
81 C.F.DunklandY.XuOrthogonalPolynomialsofSeveralVariables
82 J.P.MayberryTheFoundationsofMathematicsintheTheoryofSets
83 C.Foias,O.Manley,R.RosaandR.TemamNavier–StokesEquationsandTurbulence
84 B.PolsterandG.SteinkeGeometriesonSurfaces
85 R.B.ParisandD.KaminskiAsymptoticsandMellin–BarnesIntegrals
86 R.McElieceTheTheoryofInformationandCoding,2ndedn
87 B.A.MagurnAnAlgebraicIntroductiontoK-Theory
88 T.MoraSolvingPolynomialEquationSystemsI
89 K.BichtelerStochasticIntegrationwithJumps
90 M.LothaireAlgebraicCombinatoricsonWords
91 A.A.IvanovandS.V.ShpectorovGeometryofSporadicGroupsII
92 P.McMullenandE.SchulteAbstractRegularPolytopes
93 G.Gierzetal.ContinuousLatticesandDomains
94 S.R.FinchMathematicalConstants
95 Y.JabriTheMountainPassTheorem
96 G.GasperandM.RahmanBasicHypergeometricSeries,2ndedn
97 M.C.PedicchioandW.Tholen(eds.)CategoricalFoundations
98 M.E.H.IsmailClassicalandQuantumOrthogonalPolynomialsinOneVariable
99 T.MoraSolvingPolynomialEquationSystemsII
100 E.OlivieriandM.EuláliaVaresLargeDeviationsandMetastability
101 A.Kushner,V.LychaginandV.RubtsovContactGeometryandNonlinearDifferentialEquations
102 L.W.BeinekeandR.J.Wilson(eds.)withP.J.CameronTopicsinAlgebraicGraphTheory
103 O.StaffansWell-PosedLinearSystems
104 J.M.Lewis,S.LakshmivarahanandS.K.DhallDynamicDataAssimilation
105 M.LothaireAppliedCombinatoricsonWords
106 A.MarkoeAnalyticTomography
107 P.A.MartinMultipleScattering
108 R.A.BrualdiCombinatorialMatrixClasses
109 J.M.BorweinandJ.D.VanderwerffConvexFunctions
110 M.-J.LaiandL.L.SchumakerSplineFunctionsonTriangulations
111 R.T.CurtisSymmetricGenerationofGroups
112 H.Salzmann,T.Grundhöfer,H.HählandR.LöwenTheClassicalFields
113 S.PeszatandJ.ZabczykStochasticPartialDifferentialEquationswithLévyNoise
114 J.BeckCombinatorialGames
115 L.BarreiraandY.PesinNonuniformHyperbolicity
116 D.Z.ArovandH.DymJ-ContractiveMatrixValuedFunctionsandRelatedTopics
117 R.Glowinski,J.-L.LionsandJ.HeExactandApproximateControllabilityforDistributedParameterSystems
118 A.A.BorovkovandK.A.BorovkovAsymptoticAnalysisofRandomWalks
119 M.DezaandM.DutourSikiric´GeometryofChemicalGraphs
120 T.NishiuraAbsoluteMeasurableSpaces
121 M.PrestPurity,SpectraandLocalisation
122 S.KhrushchevOrthogonalPolynomialsandContinuedFractions:FromEuler’sPointofView
123 H.NagamochiandT.IbarakiAlgorithmicAspectsofGraphConnectivity
124 F.W.KingHilbertTransformsI
125 F.W.KingHilbertTransformsII
126 O.CalinandD.-C.ChangSub-RiemannianGeometry
127 M.Grabisch,J.-L.Marichal,R.MesiarandE.PapAggregationFunctions
128 L.W.BeinekeandR.J.Wilson(eds)withJ.L.GrossandT.W.TuckerTopicsinTopologicalGraphTheory
129 J.Berstel,D.PerrinandC.ReutenauerCodesandAutomata
130 T.G.FaticoniModulesoverEndomorphismRings
Convex Functions: Constructions,
Characterizations and Counterexamples
JonathanM.Borwein
UniversityofNewcastle,
NewSouthWales
JonD.Vanderwerff
LaSierraUniversity,
California
CAMBRIDGEUNIVERSITYPRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,
SãoPaulo,Delhi,Dubai,Tokyo
CambridgeUniversityPress
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©J.M.BorweinandJ.D.Vanderwerff2010
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Firstpublished2010
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
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ISBN978-0-521-85005-6Hardback
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Contents
Preface pageix
1 Whyconvex? 1
1.1 Why‘convex’? 1
1.2 Basicprinciples 2
1.3 Somemathematicalillustrations 8
1.4 Somemoreappliedexamples 10
2 ConvexfunctionsonEuclideanspaces 18
2.1 Continuityandsubdifferentials 18
2.2 Differentiability 34
2.3 ConjugatefunctionsandFenchelduality 44
2.4 Furtherapplicationsofconjugacy 64
2.5 Differentiabilityinmeasureandcategory 77
2.6 Second-orderdifferentiability 83
2.7 Supportandextremalstructure 91
3 FinerstructureofEuclideanspaces 94
3.1 Polyhedralconvexsetsandfunctions 94
3.2 Functionsofeigenvalues 99
3.3 Linearandsemidefiniteprogrammingduality 107
3.4 Selectionsandfixedpoints 111
3.5 Intotheinfinite 117
4 ConvexfunctionsonBanachspaces 126
4.1 Continuityandsubdifferentials 126
4.2 Differentiabilityofconvexfunctions 149
4.3 Variationalprinciples 161
4.4 ConjugatefunctionsandFenchelduality 171
4.5 Cˇebyševsetsandproximality 186
4.6 Smallsetsanddifferentiability 194
5 Dualitybetweensmoothnessandstrictconvexity 209
5.1 Renorming:anoverview 209
5.2 Exposedpointsofconvexfunctions 232
5.3 Strictlyconvexfunctions 238
5.4 Moduliofsmoothnessandrotundity 252
5.5 Lipschitzsmoothness 267
viii Contents
6 Furtheranalytictopics 276
6.1 Multifunctionsandmonotoneoperators 276
6.2 Epigraphicalconvergence:anintroduction 285
6.3 Convexintegralfunctionals 301
6.4 Stronglyrotundfunctions 306
6.5 Traceclassconvexspectralfunctions 312
6.6 Deepersupportstructure 317
6.7 Convexfunctionsonnormedlattices 329
7 BarriersandLegendrefunctions 338
7.1 Essentialsmoothnessandessentialstrictconvexity 338
7.2 Preliminarylocalboundednessresults 339
7.3 Legendrefunctions 343
7.4 ConstructionsofLegendrefunctionsinEuclideanspace 348
7.5 FurtherexamplesofLegendrefunctions 353
7.6 ZoneconsistencyofLegendrefunctions 358
7.7 Banachspaceconstructions 368
8 ConvexfunctionsandclassificationsofBanachspaces 377
8.1 Canonicalexamplesofconvexfunctions 377
8.2 Characterizationsofvariousclassesofspaces 382
8.3 Extensionsofconvexfunctions 392
8.4 Someothergeneralizationsandequivalences 400
9 MonotoneoperatorsandtheFitzpatrickfunction 403
9.1 Monotoneoperatorsandconvexfunctions 403
9.2 Cyclicandacyclicmonotoneoperators 413
9.3 MaximalityinreflexiveBanachspace 433
9.4 Furtherapplications 439
9.5 Limitingexamplesandconstructions 445
9.6 ThesumtheoremingeneralBanachspace 449
9.7 Moreaboutoperatorsoftype(NI) 450
10 Furtherremarksandnotes 460
10.1 Backtothefinite 460
10.2 Notesonearlierchapters 470
Listofsymbols 483
References 485
Index 508
Preface
Thisbookonconvexfunctionsemergesoutof15yearsofcollaborationbetweenthe
authors.Itisfarfrombeingthefirstonthesubjectnorwillitbethelast.Itisneither
abookon convexanalysissuchasRockafellar’sfoundational1970book[369]nor
abookon convexprogramming suchasBoydandVandenberghe’sexcellentrecent
text [128]. There are a number of fine books – both recent and less so – on both
thosesubjectsoron convexity andrelatedlyon variationalanalysis. Bookssuchas
[371,255,378,256,121,96,323,332]complementoroverlapinvariouswayswith
our own focus which is to explore the interplay between the structure of a normed
spaceandthepropertiesofconvexfunctionswhichcanexistthereon.Insomeways,
amongthemostsimilarbookstooursarethoseofPhelps[349]andofGiles[229]in
thatbothalsostraddlethefieldsofgeometricfunctionalanalysisandconvexanalysis
–butwithouttheconvexfunctionitselfbeingthecentralcharacter.
Wehavestructuredthisbooksoastoaccommodateavarietyofreaders.Thisleads
tosomeintentionalrepetition.Chapter1makesthecasefortheubiquityofconvexity,
largelybywayofexamples,manybutnotallofwhicharefollowedupinlaterchapters.
Chapter2thenprovidesafoundationforthestudyofconvexfunctionsinEuclidean
(finite-dimensional)space,andChapter3reprisesimportantspecialstructuressuch
aspolyhedrality,eigenvalueoptimizationandsemidefiniteprogramming.
Chapters 4 and 5 play the same role in (infinite-dimensional) Banach space.
Chapter 6 comprises a number of other basic topics such as Banach space selec-
tion theorems, set convergence, integral functionals, trace-class spectral functions
andfunctionsonnormedlattices.
Theremainingthreechapterscanbereadindependentlyofeachother.Chapter7
examinesthestructureof Legendrefunctionswhichcomprisesthosebarrierfunctions
whichareessentiallysmoothandessentiallystrictlyconvexandconsidershowthe
existenceofsuchbarrierfunctionsisrelatedtothegeometryoftheunderlyingBanach
space; as always the nicer the space (e.g. is it reflexive, Hilbert or Euclidean?) the
morethatcanbeachieved.Thiscouplingbetweenthespaceandtheconvexfunctions
whichmaysurviveonitisattackedmoremethodicallyinChapter8.
Chapter9investigates(maximal)monotoneoperatorsthroughtheuseofaspecial-
izedclassofconvexrepresentativefunctionsofwhichtheFitzpatrickfunctionisthe
progenitor.Wehavewrittenthischaptersoastomakeitmoreusableasastand-alone
sourceonconvexityanditsapplicationstomonotoneoperators.
