Table Of ContentGrundlehren 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
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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, con-
vexity,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 prop-
erties, 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
ofCarathe´odory,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
