Lectures on Discrete and Polyhedral Geometry

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1、LecturesonDiscreteandPolyhedralGeometryIgorPakJune5,2008ContentsIntroduction3Acknowledgments7Basicdefinitionsandnotations8PartI.Basicdiscretegeometry1.TheHellytheorem112.Carath´eodoryandB´ar´anytheorems203.TheBorsukconjecture264.Fairdivision325.Inscribedandcircumscri

2、bedpolygons396.DysonandKakutanitheorems577.Geometricinequalities648.Combinatoricsofconvexpolytopes759.Centerofmass,billiardsandthevariationalprinciple8510.Geodesicsandquasi-geodesics9711.TheSteinitztheoremanditsextensions10312.Universalityofpointandlineconfigurations

3、11013.Universalityoflinkages11914.Triangulations12615.Hilbert’sthirdproblem13916.Polytopealgebra14717.Dissectionsandvaluations15818.Mongeproblemforpolytopes16819.Regularpolytopes17520.Kissingnumbers18312PartII.Discretegeometryofcurvesandsurfaces21.Thefourvertextheor

4、em19122.Relativegeometryofconvexpolygons20523.Globalinvariantsofcurves21424.Geometryofspacecurves22725.Geometryofconvexpolyhedra:basicresults23926.Cauchytheorem:thestatement,theproofandthestory24727.Cauchytheorem:extensionsandgeneralizations25528.MeancurvatureandPog

5、orelov’slemma26229.Sen’kin-Zalgaller’sproofoftheCauchytheorem27230.Flexiblepolyhedra27831.Thealgebraicapproach28532.Staticrigidity29133.Infinitesimalrigidity30134.Proofofthebellowsconjecture30635.TheAlexandrovcurvaturetheorem31436.TheMinkowskitheorem32137.TheAlexandr

6、ovexistencetheorem32838.Bendablesurfaces33539.Volumechangeunderbending33940.Foldingsandunfoldings347PartIII.Details,details...41.Appendix35941.1.Theareaofsphericalpolygons35941.2.Thelawofcosinesforsphericaltriangles35941.3.Theirrationalityof(arccos1)/π360341.4.Ratio

7、nalapproximations36041.5.TheMinkowskiinequality36141.6.TheequalitypartintheBrunn–Minkowskiinequality36141.7.TheCayley–Mengerdeterminant36341.8.Thetheoryofplaces36541.9.Themappinglemma36742.Additionalproblemsandexercises36942.1.Problemsonpolygonsandpolyhedra36942.2.V

8、olume,areaandlengthproblems37142.3.Miscellaneousproblems374Hints,solutionsandreferencestoselectedexercises376References408Notationforthere