离散几何中的研究问题

0.DefinitionsandNotations

1.DensityProblemsforPackingsandCoverings

1.1BasicQuestionsandDefinitions

1.2TheLeastEconomicalConvexSetsforPacking

1.3TheLeastEconomicalConvexSetsforCovering

1.4HowEconomicalAretheLatticeArrangements?

1.5PackingwithSemidisks,andtheRoleofSymmetry

1.6PackingEqualCirclesintoSquares,Circles,Spheres

1.7PackingEqualCirclesorSquaresintrip

1.8TheDensestPackingofSpheres

1.9TheDensestPackingsofSpecificConvexBodies

1.10LinkingPackingandCoveringDensities

1.11SausageProblemsandCatastrophes

2.StructuralPackingandCoveringProblems

2.1DecompositionofMultiplePackingsandCoverings

2.2SolidandSaturatedPackingsandReducedCoverings

2.3StablePackingsandCoverings

2.4KissingandNeighborlyConvexBodies

2.5ThinPackingswithManyNeighbors

2.6PermeabilityandBlockingLightRays

3.PackingandCoveringwithHomotheticCopies

3.1PotatoBagProblems

3.2overingonvexBodywithItsHomotheticCopies

3.3Levi-HadwigerCoveringProblemandIllumination

3.4CoveringallbySlabs

3.5PointTrappingandImpassableLatticeArrangements

4．TilingProblems

4．1TilingthePlanewithCongruentRegions

4．2AperiodicTilingsandTilingswithFivefoldSymmetry

4．3TilingSpacewithPolytopes

5．DistanceProblems

5．1TheMaximumNumberofUnitDistancesintheP1ane

5．2TheNumberofEqualDistancesinOtherSpaces

5．3TheMinimumNumberofDistinctDistancesintheP1an

5．4TheNumberofDistinctDistancesinOtherSpaces

5．5RepeatedDistancesinPointSetsinGeneralPosition

5．6RepeatedDistancesinPointSetsinConvexPosition

5．7FrequentSmallDistancesandTouchingPairs

5．8FrequentLargeDistances

5．9ChromaticNumberofUnit—DistanceGraphs

5．10FurtherProblemsonRepeatedDistances

5．11IntegralorRationalDistances

6．ProblemsonRepeatedSubconfigurations

6．1RepeatedSimplicesandOtherPatterns

6．2RepeatedDirections，Angles，Areas

6．3EuclideanRamseyProblems．

7．IncidenceandArrangementProblems

7．1TheMaximumNumberofIncidences

7．2Sylvester—Gallai—TypeProblems

7．3LineArrangementsSpannedbyointSet8．ProblemsonPointsinGeneralPosition

8．1StructureoftheSpaceofOrderTypes

8．2ConvexPolygonsandtheErd6s—SzekeresProblem

8．3HalvingLinesandRelatedProblems

8．4ExtremalNumberofSpecialSubconfigurations

8．5OtherProblemsonPointsinGeneralPosition

9．GraphDrawingsandGeometricGraphs

9．1GraphDrawings

9．2DrawingPlanarGraphs

9．3TheCrossingNumber

9．4OtherCrossingNumbers

9．5FromThracklestoForbiddenGeometricSubgraphs

9．6FurtherTurin-TypeProblems

9．7Ramsey—TypeProblems

9．8GeometricHypergraphs

10．LatticePointProblems

10．1PackingLatticePointsinSubspaces

10．2CoveringLatticePointsbySubspaces

10．3SetsofLatticePointsAvoidingOtherRegularities

10．4VisibilityProblemsforLatticePoints

11．GeometricInequalities

11．1IsoperimetricInequalitiesforPolygonsandPolytopes

11．2Heilbronn-TypeProblems

11．3CircumscribedandInscribedConvexSets

11．4UniversalCovers

11．5ApproximationProblems

12．Index

12．1AuthorIndex

12．2SubjectIndex

　　本书是由LeoMoser牵头，花赞25年著成，书中包括500余个颇具吸引力的公开问题，理解其中许多问题并不需要太多的准备知识。书中的各章很大程度上内容自含，概述了离散几何，介绍了各个问题的历史细节及最重要的相关结果。本书可作为参考书，供致力数学研究，热爱美妙数学问题并不遗余力地试图加以解决的那些专业数学家和研究生查阅。