积分几何与几何概率
2009-5
世界图书出版公司
路易斯桑塔洛
404
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This monograph is the first in a projected series on Probability Theory. Though its title "Integral Geometry" may appear somewhat unusual in thiscontext it is nevertheless quite appropriate, for Integral Geometry is anoutgrowth of what in the olden days was referred to as "geometric probabil-ities." Originating, as legend has it, with the Buffon needle problem （which afternearly two centuries has lost little of its elegance and appeal）, geometricprobabilities have run into difficulties culminating in the paradoxes ofBertrand which threatened the fledgling field with banishment from the homeof Mathematics. In rescuing it from this fate, Poincar6 made the suggestionthat the arbitrariness of definition underlying the paradoxes could be removedby tying closer the definition of probability with a geometric group of which itwould have to be an invariant. Thus a union of concepts was born that was to become Integral Geometry. It is unfortunate that in the past forty or so years during which ProbabilityTheory experienced its most spectacular rise to mathematical prominence,Integral Geometry has stayed on its fringes. Only quite recently has there beena reawakening of interest among practitioners of Probability Theory in thisbeautiful and fascinating branch of Mathematics, and thus the book byProfessor Santal6, for many years the undisputed leader in the field of IntegralGeometry, comes at a most appropriate time. Complete and scholarly, the book also repeatedly belies the popular beliefthat applicability and elegance are incompatible. Above all the book should remind all of us that Probability Theory ismeasure theory with a "soul" which in this case is provided not by Physics or bygames of chance or by Economics but by the most ancient and noble of allof mathematical disciplines, namely Geometry.
Though its title "Integral Geometry" may appear somewhat unusual in thiscontext it is nevertheless quite appropriate, for Integral Geometry is anoutgrowth of what in the olden days was referred to as "geometric probabil-ities." Originating, as legend has it, with the Buffon needle problem （which afternearly two centuries has lost little of its elegance and appeal）, geometricprobabilities have run into difficulties culminating in the paradoxes ofBertrand which threatened the fledgling field with banishment from the homeof Mathematics. In rescuing it from this fate, Poincar6 made the suggestionthat the arbitrariness of definition underlying the paradoxes could be removedby tying closer the definition of probability with a geometric group of which itwould have to be an invariant.
Editor's StatementForewordPrefacePart Ⅰ.INTEGRAL GEOMETRY IN THE PLANE Chapter 1.Convex Sets in the Plane Chapter 2.Sets of Points and Poisson Processes in the Plane Chapter 3.Sets of Lines in the Plane Chatper 4.Pairs of Points and Pairs of Lines Chapter 5.Sets of Strips in the Plane Chapter 6.The Group of Motions in the Plane:Kinematic Density Chapter 7.Fundamental Formulas of Poincare and Blaschke Chapter 8.Lattices of FiguresPart Ⅱ.GENERAL INTEGRAL GEOMETRY Chapter 9.Differential Forms and Lie Groups Chapter 10.Density and Measure in Homogeneous Spaces Chapter 11.The Affine Groups Chpater 12.The Group of Motions in EnPart Ⅲ.INTEGRAL GEOMETRY IN En Chapter 13.Convex Sets in En Chapter 14.Linear Subspaces,Convex Sets,and Compact Manifolds Chapter 15.The Kinematic Density in En Chpater 16.Geometric and Statistical Applications; StereologyPart Ⅳ.INTEGRAL GEOMETRY IN SPACES OF CONSTANT CURVATURE Chapter 17.Noneuclidean Integral Geometry Chapter 18.Crofton's Formulas and the Kinematic FundaⅠmental Formula in Noneuclidean Spaces Chapter 19.Integral Geometry and Foliated Spaces; Trends in Integral GeometryAppendix.Differential Forms and Exterior CalculusBibliography and ReferencesAuthor IndexSubject Index
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不错，很易读的书，很好