Lectures on Sphere Arrangements - the Discrete Geometric Side

186 Pages·2013·3.06 MB·English

by  Károly Bezdek

Fields Institute Monographs 32
The Fields Institute for Research in Mathematical Sciences
Károly Bezdek
Lectures on Sphere
Arrangements –
the Discrete
Geometric Side
Fields Institute Monographs
VOLUME 32
The Fields Institute for Research in Mathematical Sciences
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Ka´roly Bezdek
Lectures on Sphere
Arrangements—the Discrete
Geometric Side
123
TheFieldsInstituteforResearch
intheMathematicalSciences
Ka´rolyBezdek
DepartmentofMathematicsandStatistics
UniversityofCalgary
Calgary,AB,Canada
ISSN1069-5273 ISSN2194-3079(electronic)
ISBN978-1-4614-8117-1 ISBN978-1-4614-8118-8(eBook)
DOI10.1007/978-1-4614-8118-8
SpringerNewYorkHeidelbergDordrechtLondon
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Mathematics Subject Classification (2010): 52-02, 52C07, 52C10, 52C17, 52C22, 52C25, 52C35,
52C45,52C99,52B10,52B11,52B60,52B99,52A05,52A21,52A38,52A40,52A55,52A99
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Preface
Thethematicprogramon“DiscreteGeometryandApplications”tookplaceatthe
Fields Institute for Research in Mathematical Sciences in Toronto between July 1
and December 31, 2011.The core partof the bookis based on my three lectures,
deliveredatthetheFieldsInstituteunderthetitles“Contactnumbersforcongruent
spherepackings”(September23,2011),“Rigidball-polyhedra”(October11,2011),
and“OnastrongversionoftheKeplerconjecture”(November17,2011).
Onecanbrieflydescribediscretegeometryasthestudyofdiscretearrangements
of geometric objects in Euclidean, as well as in non-Euclidean spaces. J. Kepler
was the first to raise discrete geometryproblemson packingsof balls in the early
1610s,butthesystematicresearchbeganinthelate1940swiththeworkofL.Fejes
To´th. The Hungarianschoolhe foundedfocused mainly on packing and covering
problems,whileanumberofgreatmathematicianshelpedtolayabroadfoundation
for the emerging field of discrete geometry, including H. S. M. Coxeter, J. H.
Conway,B.N.Delaunay,B.Gru¨nbaum,V.Klee,andC.A.Rogers.Twoactiveareas
that have been outstanding from the birth of discrete geometry are dense sphere
packingsandtilings.Bothoccupyasubstantialpartofmybookaswell.Thereisa
chapter on unit sphere packingsand there are a number of sections that apply the
methodof Delaunayand Voronoitilingsin the solutionsof a varietyof problems.
Spherepackingsandtilingshaveaverystrongconnectiontonumbertheory,coding,
groups, and mathematical programming.In particular, the latter greatly helped to
achieve recent breakthroughresults. Extending the tradition of studying packings
ofspheres, anothercore topicof mybookis the investigationof the monotonicity
ofvolumeundercontractionsofarbitraryarrangementsofspheres.Theresearchon
thisfundamentaltopic startedwith the conjectureofE. T.Poulsen andM.Kneser
inthelate1950s.Thethirdmajortopicofmybookcanbefoundunderthesections
onball-polyhedraintroducinganextensionofthetheoryofconvexpolyhedralsets
tothefamilyofintersectionsofcongruentballs.Thispartofmybookisconnected
inmanywaystotheabovementionedmajortopicsanditisalsoconnectedtosome
other important research areas as well including the one on coverings by planks
(withclosetiestogeometricanalysis).Thistopicistheforthmajoroneinmybook
vii
viii Preface
discussedundercoveringsbycylinders.Theresearchworkonthelattertopicstarted
withaconjectureofA.Tarskiintheearly1930s1.
My book is aimed at advanced undergraduate and early graduate students, as
well as interested researchers. In addition to leading the reader to the frontiers
of geometric research on sphere arrangements,it gives a short introductionto the
relevant modern parts of discrete geometry. I have structured the book in such
a way that the four major research topics (unit sphere packings, contractions of
sphere arrangements, ball-polyhedra,and coveringsby cylinders) are surveyed in
individual chapters (Chaps.1, 3, 5, and 7) each followed by a chapter with a
collectionofselectedproofs(Chaps.2,4,6,and8).Thesurveychaptersarereadable
independentlyfromeachother.Theselectedproofscombineelementaryandconvex
geometrywithanalyticandinsomecases,probabilisticortopologicalideas.They
are the results of the author’s joint work with a number of discrete geometers.
In addition, an independently understandable collection of unsolved problems is
compiledinChap.9.
I am very much indebted to all my students and colleagues who attended my
lecturesandactivelyparticipatedinthediscussionsattheFieldsInstituteinthefall
of 2011. Furthermore,I want to thank the support of a number of colleagues and
friendsinparticular,TedBisztriczky(Univ.ofCalgary,Canada),Ka´rolyBo¨ro¨czky
(Eo¨tvo¨s Univ., Hungary), Robert Connelly (Cornell Univ., USA), Bala´zs Csiko´s
(Eo¨tvo¨s Univ., Hungary), Antoine Deza (McMaster Univ., Canada), Ga´bor Fejes
To´th (Re´nyi Inst., Hungary), Herbert Edelsbrunner (Duke Univ., USA and IST,
Austria), Ferenc Friedler (Univ. of Pannonia, Hungary), Thomas C. Hales (Univ.
of Pittsburgh, USA), Ja´nos Pach (EPFL, Switzerland and Re´nyi Inst., Hungary),
KonradSwanepoel(LSE,UK),SalvatoreTorquato(PrincetonUniv.,USA),AsiaI.
Weiss(YorkUniv.,Canada),andYinyuYe(StanfordUniv.,USA).Itisaparticular
pleasure for me to acknowledge my long-lasting research collaboration with my
brother Andra´s Bezdek (Auburn Univ., USA and Re´nyi Inst., Hungary) as well
as with my friendsTed Bisztriczky (Univ.of Calgary,Canada) and Bob Connelly
(Cornell Univ., USA). Also, it is a pleasure to acknowledge the excellent support
providedby the Fields Institute; in particular, I would like to offer special thanks
to Edward Bierstone, Alison Conway, Claire Dunlop, Matheus Grasselli, Debbie
Iscoe,MatthiasNeufang,andCarl Riehm.Also, specialthanksare dueto Samuel
Reid (Univ. of Calgary, Canada) for the expressive drawings. Last but not least, I
wish to thank my three sons, Da´niel, Ma´te´, and Ma´rk, and in particular,my wife,
E´va, whose strong support and encouragementhelped me a great deal during the
longhoursofwriting.
Calgary,AB,Canada Ka´rolyBezdek,CanadaResearchChair
1TheauthorwassupportedbytheCanadaResearchChairprogramaswellasaNaturalSciences
andEngineeringResearchCouncilofCanadaDiscoveryGrant.
Contents
1 UnitSpherePackings........................................................ 1
1.1 TheContactNumberProblemofFiniteSpherePackings........... 1
1.2 LowerBoundsforVoronoiCellsinSpherePackings................ 5
1.3 DenseSpherePackingsinEuclidean3-Space........................ 9
1.4 OnSpherePackingsinHighDimensions ............................ 12
1.5 UniformlyStableandPeriodicExtremeLattices .................... 14
2 ProofsonUnitSpherePackings............................................ 17
2.1 ProofofTheorem1.1.6................................................ 17
2.1.1 AnUpperBoundforSpherePackings:Proofof(i)........ 18
2.1.2 AnUpperBoundforthefccLattice:Proofof(ii).......... 23
2.1.3 OctahedralUnitSpherePackings:Proofof(iii)............ 25
2.2 ProofofTheorem1.2.4................................................ 26
2.2.1 TheVoronoiStarofVoronoiCells.......................... 26
2.2.2 EstimatingtheVolumeofaVoronoiStarfromBelow..... 27
2.3 ProofofTheorem1.2.5................................................ 28
2.3.1 MetricPropertiesofVoronoiCells.......................... 28
2.3.2 WedgesofTypesI,II,andIII,andTruncatedWedges..... 29
2.3.3 TheLemmaofComparison ................................. 32
2.3.4 VolumeFormulasfor(Truncated)Wedges ................. 34
2.3.5 TheIntegralRepresentationofSurfaceDensity............ 34
2.3.6 TruncationofWedgesIncreasestheSurfaceDensity...... 37
2.3.7 Maximum Surface Density in Truncated
WedgesofTypeI............................................. 38
2.3.8 SurfaceDensityinTruncatedWedgesofTypeII........... 41
2.3.9 The OverallEstimate of Surface Density
inVoronoiCells.............................................. 42
2.4 ProofofTheorem1.3.3................................................ 43
2.4.1 AverageSurface Areaof Cells in Normal
TilingsofaCube............................................. 43
2.4.2 AverageSurfaceAreaofCellsinNormalTilings.......... 45
ix