Weil W. A course on convex geometry (lecture notes).pdf

ACourse

ConvexGeometry

WolfgangWeil

UniversityofKarlsruhe

2002/2003

revisedversion2004/2005

Preface

ThefollowingnoteswerewrittenbeforeandduringthecourseonConvexGeom-

etrywhichwasheldattheUniversityofKarlsruheinthewinterterm2002/2003.

AlthoughthiswastheﬁrstcourseonthistopicwhichwasgiveninEnglish,the

materialpresentedwasbasedonpreviouscoursesinGermanwhichhavebeengiven

severaltimes,mostlyinsummerterms.Incomparisonwiththesepreviouscourses,

thestandardprogramwascomplementedbysectionsonsurfaceareameasuresand

projectionfunctionsaswellasbyashortchapteronintegralgeometricformulas.

TheideaherewastolaythebasisforthecourseonStochasticGeometrywhichwill

followinthesummerterm.

Theexercisesattheendofeachsectioncontainalltheweeklyproblemswhich

werehandedoutduringthecourseanddiscussedintheweaklyexercisesession.

Moreover,Ihaveincludedafewadditionalexercises(someofwhicharemoredif-

ﬁcult)andevensomesevereorevenunsolvedproblems.Thelistofexercisesand

problemsisfarfrombeingcomplete,infactthenumberdecreasesinthelatersec-

tionsduetothelackoftimewhilepreparingthesenotes.

IthankMatthiasHevelingandMarkusKiderlenforreadingthemanuscriptand

givinghintsforcorrectionsandimprovements.

Karlsruhe,February2003 WolfgangWeil

Duringarepetitionofthecoursein2003/2004anumberofmisprintsandsmall

errorshavebeendetected.Theyarecorrectedinthecurrentversion.Also,afew

remarksandfurtherexerciseshavebeenadded.

Karlsruhe,October2004 WolfgangWeil

Contents

Introduction 7

Preliminariesandnotations 9

1Convexsets 11

1.1Algebraicproperties...........................11

1.2Combinatorialproperties.........................18

1.3Topologicalproperties..........................22

1.4Supportandseparationtheorems....................27

1.5Extremalrepresentations.........................33

2Convexfunctions 39

2.1Propertiesandoperationsofconvexfunctions.............39

2.2Regularityofconvexfunctions......................47

2.3Thesupportfunction...........................52

3Convexbodies 59

3.1Thespaceofconvexbodies........................59

3.2Volumeandsurfacearea.........................68

3.3Mixedvolumes..............................73

3.4TheBrunn-MinkowskiTheorem...................86

3.5Surfaceareameasures..........................93

3.6Projectionfunctions...........................107

4Integralgeometricformulas 115

4.1Invariantmeasures............................115

4.2Projectionformulas............................122

4.3Sectionformulas..............................124

Bibliography 131

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