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Grundlehren der 336 mathematischen Wissenschaften ASeriesofComprehensiveStudies inMathematics Serieseditors M.Berger B.Eckmann P.delaHarpe F.Hirzebruch N.Hitchin L.Hörmander M.-A.Knus A.Kupiainen G. Lebeau M.Ratner D.Serre Ya.G.Sinai N.J.A.Sloane A. Vershik M.Waldschmidt Editor-in-Chief A.Chenciner J.Coates S.R.S.Varadhan Peter M. Gruber Convex and Discrete Geometry ABC Peter M. Gruber Institute of Discrete Mathematics and Geometry Vienna University of Technology Wiedner Hauptstrasse 8-10 1040 Vienna, Austria e-mail: [email protected] LibraryofCongressControlNumber:2007922936 MathematicsSubjectClassification(2000): Primary:52-XX, 11HXX Secondary:05Bxx,05Cxx,11Jxx, 28Axx, 46Bxx, 49Jxx, 51Mxx, 53Axx, 65Dxx, 90Cxx, 91Bxx ISSN0072-7830 ISBN 978-3-540-71132-2SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2007 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesettingbytheauthorand SPiusingaSpringerLATEXmacropackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:11966739 41/SPi 543210 Preface Inthisbookwegiveanoverviewofmajorresults,methodsandideasofconvexand discretegeometryandtheirapplications.Besidesbeingagraduate-levelintroduction tothefield,thebookisapracticalsourceofinformationandorientationforconvex geometers.Itshouldalsobeofusetopeopleworkinginotherareasofmathematics andintheappliedfields. Wehopetoconvincethereaderthatconvexityisoneofthosehappynotionsof mathematicswhich,likegroupormeasure,satisfyagenuinedemand,aresufficiently generaltoapplytonumeroussituationsand,atthesametime,sufficientlyspecialto admitinteresting,non-trivialresults.Itisouraimtopresentconvexityasabranchof mathematicswithamultitudeofrelationstootherareas. Convexgeometrydatesbacktoantiquity.Resultsandhintstoproblemswhichare ofinteresteventodaycanalreadybefoundintheworksofArchimedes,Euclidand Zenodorus. We mention the Platonic solids, the isoperimetric problem, rigidity of polytopalconvexsurfacesandtheproblemofthevolumeofpyramidsasexamples. Contributionstoconvexityinmoderntimesstartedwiththegeometricandanalytic workofGalileo,theBernoullis,CauchyandSteinerontheproblemsfromantiquity. These problems were solved only in the nineteenth and early twentieth century by Cauchy,SchwarzandDehn.ResultswithoutantecedentsinantiquityincludeEuler’s polytopeformulaandBrunn’sinequality.Muchofmodernconvexitycameintobeing withMinkowski.ImportantlatercontributorsareBlaschke,Hadwiger,Alexandrov, Pogorelov, and Klee, Groemer, Schneider, McMullen together with many further living mathematicians. Modern aspects of the subject include surface and curva- ture measures, the local theory of normed spaces, best and random approximation, affine-geometric features, valuations, combinatorial and algebraic polytope theory, algorithmicandcomplexityproblems. Keplerwasthefirsttoconsiderproblemsofdiscretegeometry,inparticularpack- ingofballsandtiling.HisworkwascontinuedbyThue,butthesystematicresearch began with Fejes To´th in the late 1940s. The Hungarian school deals mainly with packingandcoveringproblems.Amongstnumerousothercontributorswemention Rogers,PenroseandSloane.Tilingproblemsareaclassicalandalsoamoderntopic. The ball packing problem with its connections to number theory, coding and the VI Preface theoryoffinitegroupsalwayswasandstillisofgreatimportance.Morerecentisthe researchonarrangements,matroidsandtherelationstographtheory. Thegeometryofnumbers,theelegantoldersisterofdiscretegeometry,wascre- atedbyLagrange,Gauss,Korkin,Zolotarev,Fedorov,theleadingfiguresMinkowski andVorono˘ı,byBlichfeldt,byDelone,RyshkovandtheRussianschool,bySiegel, Hlawka,Schmidt,Davenport,Mahler,Rogersandothers.Centralproblemsofmod- ernresearcharethetheoryofpositivedefinitequadraticforms,includingreduction, algorithmicquestionsandthelatticeballpackingproblem. From tiny branches of geometry and number theory a hundred years ago, convexity,discretegeometryandgeometryofnumbersdevelopedintowell-established areasofmathematics.Nowtheirdoorsarewideopentootherpartsofmathematics and a number of applied fields. These include algebraic geometry, number theory, in particular Diophantine approximation and algebraic number theory, theta series, error correcting codes, groups, functional analysis, in particular the local theory of normedspaces,thecalculusofvariations,eigenvaluetheoryinthecontextofpartial differential equations, further areas of analysis such as geometric measure theory, potential theory, and also computational geometry, optimization and econometrics, crystallography,tomographyandmathematicalphysics. Westart withconvexity in thecontext of real functions. Then convex bodies in Euclideanspaceareinvestigated,makinguseofanalytictoolsand,insomecases,of discreteandcombinatorialideas.Next,variousaspectsofconvexpolytopesarestud- ied. Finally, we consider geometry of numbers and discrete geometry, both from a rathergeometricpointofview.Formoredetaileddescriptionsofthecontentsofthis bookseetheintroductionsoftheindividualchapters.Applicationsdealwithmeasure theory,thecalculusofvariations,complexfunctiontheory,potentialtheory,numer- ical integration, Diophantine approximation, matrices, polynomials and systems of polynomials,isoperimetricproblemsofmathematicalphysics,crystallography,data transmission,optimizationandotherareas. Whenwritingthebook,Ibecameawareofthefollowingphenomenainconvex and discrete geometry. (a) Convex functions and bodies which have certain properties, have these properties often in a particularly strong form. (b) In complicated situations, the average object has often almost extremal properties. (c) In simple situations, extremal configurations are often regular or close to regular. The reader willfindnumerousexamplesconfirmingthesestatements. In general typical, rather than refined results and proofs are presented, even if more sophisticated versions are available in the literature. For some results more thanoneproofisgiven.Thiswasdonewheneachproofshedsdifferentlightonthe problem.Toolsfromotherareasareusedfreely.Thereaderwillnotethattheproofs varyalot.Whileinthegeometryofnumbersandinsomemoreanalyticbranchesof convex geometry most proofs are crystal clear and complete, in other cases details areleftoutinordertomaketheideasoftheproofsbettervisible.Sometimesweused more intuitive arguments which, of course, can be made precise by inserting addi- tionaldetailorbyreferringtoknownresults.Thereadershouldkeepinmindthatall thisistypicalofthevariousbranchesofconvexanddiscretegeometry.Someproofs are longer than in the original literature. While most results are proved, there are Preface VII someimportanttheorems,theproofsofwhichareregrettablyomitted.Someofthe proofsgivenarecomplicatedandwesuggestskippingtheseatafirstreading.There areplentyofcomments,somestatingtheauthor’spersonalopinions.Theemphasis ismoreonthesystematicaspectofconvexitytheory.Thismeansthatmanyinterest- ingresultsarenotevenmentioned.Inseveralcaseswestudyanotioninthecontext ofconvexityinonesection(e.g.Jordanmeasure)andapplyadditionalpropertiesof itinanothersection(e.g.Fubini’stheorem).Wehavetriedtomaketheattributions tothebestofourknowledge,butthehistoryofconvexitywouldformacomplicated book.Inspiteofthis,historicalremarksandquotationsaredispersedthroughoutthe book. The selection of the material shows the author’s view of convexity. Several sub-branches of convex and discrete geometry are not touched at all, for example axiomatic and abstract convexity, arrangements and matroids, and finite packings, othersarebarelymentioned. Istartedtoworkinthegeometryofnumbersasastudentandbecamefascinated byconvexanddiscretegeometryslightlylater.Myinterestwasgreatlyincreasedby thestudyoftheseminallittlebooksofBlaschke,FejesTo´th,HadwigerandRogers, andIhavebeenworkinginthesefieldseversince. For her great help in the preparation of the manuscript I am obliged to Edith Rosta and also to Franziska Berger who produced most of the figures. Kenneth Stephenson provided the figures in the context of Thurston’s algorithm. The whole manuscript or part of it was read by Iskander Aliev, Keith Ball, Ka´roly Bo¨ro¨czky Jr.,Ga´borFejesTo´th,AugustFlorian,RichardGardner,HelmutGroemer,Christoph Haberl, Rajinder Hans-Gill, Martin Henk and his students Eva Linke and Matthias Henze, Jiˇr´ı Matousˇek, Peter McMullen, Matthias Reitzner, Rolf Schneider, Franz Schuster,TonyThompson,Jo¨rgWills,Gu¨nterZieglerandChuanmingZongandhis students.UsefulsuggestionsareduetoPeterEngel,HendrikLenstra,PeterManiand AlexanderSchrijver.Ithankallthesecolleagues,studentsandfriendsfortheirefforts incorrectingmathematicalandlinguisticerrors,forpointingoutrelevantresultsand referenceswhichIhadmissed,andfortheirexpertadvice.Thanksinparticularare duetoPaulGoodeyforhishelp. I am indebted to numerous mathematicians for discussions, side remarks, questions, correct and false conjectures, references to the literature, orthodox and, sometimes, unorthodox views, and interesting lectures over many years. The book reflects muchofthis.Mostofthesefriends,colleagues andstudentsarementioned inthebookviatheirwork.Ifinallyrememberwithgratitudemyteacher,colleague andfriendEdmundHlawkaandmylateseniorfriendsLa´szlo´ FejesTo´thandHans Zassenhaus. Special thanks go to Springer. For their expert advice and help I thank, in particular,CatrionaByrne,StefanieZoellerandJoachimHeinze. Vienna,January2007 PeterM.Gruber Contents Preface ......................................................... V ConvexFunctions ................................................ 1 1 ConvexFunctionsofOneVariable ............................ 2 1.1 Preliminaries........................................ 2 1.2 Continuity,SupportandDifferentiability................. 4 1.3 ConvexityCriteria ................................... 12 1.4 Jensen’sandOtherInequalities......................... 12 1.5 BohrandMollerup’sCharacterizationofΓ .............. 16 2 ConvexFunctionsofSeveralVariables......................... 20 2.1 Continuity, Support and First-Order Differentiability, andaHeuristicPrinciple .............................. 20 2.2 Alexandrov’sTheoremonSecond-OrderDifferentiability .. 27 2.3 AConvexityCriterion ................................ 32 2.4 AStone–WeierstrassTypeTheorem..................... 34 2.5 A Sufficient Condition of Courant and Hilbert intheCalculusofVariations ........................... 35 ConvexBodies ................................................... 39 3 ConvexSets,ConvexBodiesandConvexHulls ................. 40 3.1 BasicConceptsandSimpleProperties................... 41 3.2 AnExcursionintoCombinatorialGeometry:TheTheorems ofCaratheódory,HellyandRadon ...................... 46 3.3 Hartogs’TheoremonPowerSeries ..................... 50 4 SupportandSeparation...................................... 52 4.1 SupportHyperplanesandSupportFunctions ............. 52 4.2 SeparationandOracles ............................... 58 4.3 Lyapunov’sConvexityTheorem........................ 60 4.4 Pontryagin’sMinimumPrinciple ....................... 64 5 TheBoundaryofaConvexBody ............................. 68 5.1 SmoothandSingularBoundaryPoints,Differentiability.... 68 X Contents 5.2 ExtremePoints ...................................... 74 5.3 Birkhoff’sTheoremonDoublyStochasticMatrices ....... 76 6 MixedVolumesandQuermassintegrals ........................ 79 6.1 Minkowski Addition, Direct Sums, Hausdorff Metric, andBlaschke’sSelectionTheorem...................... 80 6.2 Minkowski’sTheoremonMixedVolumesandSteiner’s Formula............................................ 88 6.3 PropertiesofMixedVolumes .......................... 93 6.4 QuermassintegralsandIntrinsicVolumes ................ 102 7 Valuations................................................. 110 7.1 ExtensionofValuations............................... 111 7.2 ElementaryVolumeandJordanMeasure................. 118 7.3 CharacterizationofVolumeandHadwiger’sFunctional Theorem ........................................... 126 7.4 ThePrincipalKinematicFormulaofIntegralGeometry .... 134 7.5 Hadwiger’sContainmentProblem ...................... 140 8 TheBrunn–MinkowskiInequality............................. 141 8.1 TheClassicalBrunn–MinkowskiInequality .............. 142 8.2 TheBrunn–MinkowskiInequalityforNon-ConvexSets.... 146 8.3 TheClassicalIsoperimetricandtheIsodiametricInequality andGeneralizedSurfaceArea.......................... 147 8.4 Sand Piles, Capillary Surfaces and Wulff’s Theorem inCrystallography ................................... 155 8.5 ThePre´kopa–LeindlerInequalityandtheMultiplicative Brunn–MinkowskiInequality .......................... 161 8.6 General Isoperimetric Inequalities and Concentration ofMeasure ......................................... 164 9 Symmetrization ............................................ 168 9.1 SteinerSymmetrization ............................... 168 9.2 The Isodiametric, Isoperimetric, Brunn–Minkowski, Blaschke–Santalo´ andMahlerInequalities ............... 175 9.3 SchwarzSymmetrizationandRearrangementofFunctions . 178 9.4 TorsionalRigidityandMinimumPrincipalFrequency ..... 179 9.5 CentralSymmetrizationandtheRogers–Shephard Inequality .......................................... 185 10 ProblemsofMinkowskiandWeylandSomeDynamics........... 187 10.1 AreaMeasureandMinkowski’sProblem ................ 188 10.2 IntrinsicMetric,Weyl’sProblemandRigidityofConvex Surfaces............................................ 197 10.3 EvolutionofConvexSurfacesandConvexBilliards ....... 199 11 ApproximationofConvexBodiesandItsApplications ........... 202 11.1 John’sEllipsoidTheoremandBall’sReverseIsoperimetric Inequality .......................................... 203 11.2 Asymptotic Best Approximation, the Isoperimetric ProblemforPolytopes,andaHeuristicPrinciple .......... 209 Contents XI 12 SpecialConvexBodies...................................... 218 12.1 SimplicesandChoquet’sTheoremonVectorLattices...... 218 12.2 ACharacterizationofBallsbyTheirGravitationalFields... 222 12.3 Blaschke’sCharacterizationofEllipsoids andItsApplications .................................. 225 13 TheSpaceofConvexBodies ................................. 230 13.1 BaireCategories ..................................... 231 13.2 MeasuresonC?...................................... 234 13.3 OntheMetricStructureofC ........................... 236 13.4 OntheAlgebraicStructureofC ........................ 237 ConvexPolytopes ................................................ 243 14 PreliminariesandtheFaceLattice............................. 244 14.1 BasicConceptsandSimplePropertiesofConvexPolytopes. 244 14.2 ExtensiontoConvexPolyhedraandBirkhoff’sTheorem ... 247 14.3 TheFaceLattice..................................... 252 14.4 ConvexPolytopesandSimplicialComplexes ............. 257 15 CombinatorialTheoryofConvexPolytopes .................... 258 15.1 Euler’sPolytopeFormulaandItsConversebySteinitz ford =3 ........................................... 259 15.2 ShellingandEuler’sFormulaforGenerald .............. 265 15.3 Steinitz’PolytopeRepresentationTheoremford =3...... 270 15.4 Graphs,Complexes,andConvexPolytopesforGenerald .. 272 15.5 CombinatorialTypesofConvexPolytopes ............... 277 16 VolumeofPolytopesandHilbert’sThirdProblem ............... 280 16.1 ElementaryVolumeofConvexPolytopes ................ 280 16.2 Hilbert’sThirdProblem............................... 288 17 Rigidity................................................... 292 17.1 Cauchy’sRigidityTheoremforConvexPolytopalSurfaces . 292 17.2 RigidityofFrameworks............................... 297 18 TheoremsofAlexandrov,MinkowskiandLindelo¨f .............. 301 18.1 Alexandrov’sUniquenessTheoremforConvexPolytopes .. 301 18.2 Minkowski’sExistenceTheoremandSymmetryCondition . 303 18.3 The Isoperimetric Problem for Convex Polytopes andLindelo¨f’sTheorem .............................. 308 19 LatticePolytopes........................................... 310 19.1 Ehrhart’sResultsonLatticePointEnumerators ........... 310 19.2 Theorems of Pick, Reeve and Macdonald on Volume andLatticePointEnumerators ......................... 316 19.3 TheMcMullen–BernsteinTheoremonSumsofLattice Polytopes........................................... 320 19.4 TheBetke–KneserTheoremonValuations ............... 324 19.5 NewtonPolytopes:IrreducibilityofPolynomials andtheMinding–Kouchnirenko–BernsteinTheorem....... 332 XII Contents 20 LinearOptimization ........................................ 335 20.1 PreliminariesandDuality ............................. 336 20.2 TheSimplexAlgorithm............................... 339 20.3 TheEllipsoidAlgorithm .............................. 343 20.4 LatticePolyhedraandTotallyDualIntegralSystems....... 345 20.5 HilbertBasesandTotallyDualIntegralSystems .......... 348 GeometryofNumbersandAspectsofDiscreteGeometry............... 353 21 Lattices................................................... 355 21.1 BasicConceptsandPropertiesandaLinearDiophantine Equation ........................................... 356 21.2 CharacterizationofLattices............................ 359 21.3 Sub-Lattices ........................................ 361 21.4 PolarLattices ....................................... 365 22 Minkowski’sFirstFundamentalTheorem ...................... 366 22.1 TheFirstFundamentalTheorem........................ 366 22.2 Diophantine Approximation and Discriminants ofPolynomials ...................................... 370 23 SuccessiveMinima ......................................... 375 23.1 SuccessiveMinimaandMinkowski’sSecondFundamental Theorem ........................................... 376 23.2 Jarn´ık’sTransferenceTheoremandaTheoremofPerron andKhintchine ...................................... 380 24 TheMinkowski–HlawkaTheorem ............................ 385 24.1 TheMinkowski–HlawkaTheorem...................... 385 24.2 Siegel’sMeanValueTheoremandtheVarianceTheorem ofRogers–Schmidt................................... 388 25 Mahler’sSelectionTheorem ................................. 391 25.1 TopologyontheSpaceofLattices ...................... 391 25.2 Mahler’sSelectionTheorem ........................... 392 26 TheTorusGroupEd/L...................................... 395 26.1 DefinitionsandSimplePropertiesofEd/L............... 395 26.2 TheSumTheoremofMacbeath–Kneser ................. 398 26.3 Kneser’sTransferenceTheorem ........................ 403 27 SpecialProblemsintheGeometryofNumbers.................. 404 27.1 TheProductofInhomogeneousLinearFormsandDOTU Matrices............................................ 405 27.2 Mordell’sInverseProblemandtheEpsteinZeta-Function .. 408 27.3 LatticePointsinLargeConvexBodies .................. 410 28 BasisReductionandPolynomialAlgorithms.................... 411 28.1 LLL-BasisReduction................................. 411 28.2 DiophantineApproximation,theShortestandtheNearest LatticeVectorProblem ............................... 417