Topics in integral geometry

Cover
Foreword by Chern
Foreword by Hsiung
Preface
Chapter I. Basic Properties of Convex Sets
1.1 Basic Concepts
1.1.1 Convex sets and convex curves
1.1.2 Support lines and their existence
1.2 Support Functions and Width Functions
1.2.1 Generalized normal equations of lines
1.2.2 Support function and width function of a convex set
1.2.3 Convex curve as envelope of a family of lines
1.2.4 An elementary proof of the formula of perimeter
1.3 Some Special Convex Sets
1.3.1 Convex sets of constant widths
1 3.2 Parallel convex sets
1.4 Mixed Areas of Minkowski
1 4.1 Mixedd convex sets
1.4.2 Mixed areas of Minkowaki
1.5 Surface Area of the Unit Sphere and Volume of the Unit Ball
Chapter II. Measure for Sets of Geometric Elements
2.1 Measure for Sets of Points
2.1.1 Measure for sets of points
2.1.2 Remarks
2.1.3 An integral formula
2.2 Measure for Sets of Lines
2.2.1 Measure for seta or lines
2.2.2 Two corollaries
2.2.3 Other forms of density for sets of lines
2.2.4 Proof of isoperimetric inequality
2.3 Pairs of Points and Lines
2.3.1 Density of pairs of points
2.3.2 Integral for the power of chords of a convex set
2.3.3 Remarks on integrals for power of chords of a convex set
2.3.4 Inequalities for the integrals of the power of chord of a convex set
2.3.5 Density for pairs of lines
2.3.6 Crofton's formula
2.4 Division of the Plane by Random Lines
2.4.1 Division of the convex set by random lines
2.4.2 Division of plane by random lines
2 4 3 Notes on random division
2.5 Sets of Strips in the Plane
2.5.1 Density for sets of strips
2.5.2 Generalized Buffon's needle problem
2.5.3 Further generalizations
Chapter III. Fundamental Formulas of Integral Geometry in the Plane
3.1 The Group of Motions in the Plane
3.1.1 The group of motions in the plane
3.1.2 Left and right translations
3.1.3 The differential forms on M
3.2 The Kinematic Density
3.2.1 Left and right invariant 1-forms
3.2.2 The kinematic density
3.2.3 Geometrical meaning of the kinematic measure
3.2.4 Other expressions for the kinematic measure
3.3 Poincarré's Formula
3.3.1 A new expression for the kinematic density
3.3.2 Poincaré's Formula
3.4 The Fnndamental Kinematic Formula of Blaschke
3.4.1 Total curvature of a closed curve and of a plane domain
3 4.2 Fundamental kinematic formula of Blaschke
3.4.3 Some immediate consequences of Blaschke's formula
Chapter IV. Applications of Integral Geometry in the Plane
4.1 The Isoperimetric Inequality
4.1.1 The integral geometric proof of the isoperimetric inequality
4.1.2 Stronger isoperimetric inequalities
4.1.3 An upper limit for isoperimetric deficit
4.2 Conditions for One Domain to be Able to Contain Another
4.2.1 Suflicient conditions for one domain to contain another
4.2.2 Hadwiger's conditions
4.2.3 Some consequences
4.2.4 Pseudodiameter of a domain
4.3 Kinematic Measure of a Segment of Fixed Length within a Convex Domain
4.3.1 Problem
4.3.2 A formula for kinematic measure of a segment within a convex domain
4.3.3 Generalized support function and the restricted chord function
4.3.4 A formula for m(l) expressed by the generalized support fonction
4.3.5 The measure m(l) for a rectangle
4.4 Applications of m(l) to Geometric Probability
4.4.1 Laplace extension of Buffon problem
4.4.2 Applications of m(l) to generalized Buffon problem
4.4.3 Formulas of m(l) for equilateral triangle and regular hexagon
4.5 Problem Related to the Statistical Estimating of π
4.5.1 Equidistant parallel lines
4.5.2 Grid of rectangles, independence
4.5.3 Efficiency analysis
4.6 Random Convex Set in a Lattice of Parallelograms
4 6 1 Width functions of convex sets
4.6.2 Distribution of the number of intersections
4.6.3 Hitting probabilities
4.6.4 Independence
Chapter V. Foundations of Integral Geometry in Homogeneous Spaces
5.1 Differentiable Manifolds
5.5.1 Topological space
5.5.2 Topological manifolds and differentiable manifolds
5.5.3 Differentiable functions and mappings
5.2 Vector Fields on a Manifold
5.2.1 Tangent spaces and vector fields
5.2.2 The differential of a mapping between manifolds
5.2.3 Local expressions of vector fields
5.3 Differential Forms and Exterior Differentiation
5.3.1 Covector fields
5.3.2 Tensor fields
5.3.3 The exterior algebra on a manifold
5.3.4 Fxterior differentiation
5.3.5 Expression of exterior differential by the usual differential
5.4 Integral Manifolds and Pfaffian Systems
5.4.1 Integral manifolds
5.4.2 Pfaffian system
5.5 Lie Groups and Kinematic Density of a Lie Group
5.5.1 Lie group
5.5.2 Left and right translations
5.5.3 Left invariant differential forms
5.5.4 Structure equations and structure constants for a Lie group
5.5.5 Kinematic density for Lie group
5.6 Density and Measure in Homogeneous Space
5.6.1 Actions of a Lie group on a manifold, homogeneous space
5.6.2 Conditions for the existence of invariant density on G/H
5.6.3 Weil's condition
5.6.4 Normal subgroups
5.6.5 Chern's conditions
5.6.6 Stable subgroups
5.7 A Brier Review of Integral Geometry in the Plane
Chapter VI. Integral Geometry in R^n
6.1 The Group of Motions in En
6.1.1 The group of motions and its structure equations
6.1.2 Invariant volume elements of the group of motions and its subgroups
6.2 The Density of t'-Planes in E_n
6.2.1 The density of r-planes
6.2.2 Density of r-planes about a fixed q-plane
6.2.3 The volume of the Grassmann manifold G_{r,n-r}
6.2.4 Another form of the density of r-planes in E_n
6.2.5 The density of the pairs of hyperplanes
6.2.6 The density of the pairs (L_r, L{(r)}_{i+1})
6.2.7 A density formula for points and flats
6.2.8 The density of the pairs of non-intersecting flats
6.3 Convex Sets in E_n
6.3.1 Convex sets and mixed volumes
6.3.2 Quermassintegrale
6.3.3 Cauchy's formula and Steiner's formula
6.3.4 The mean value of W'_i(K'_{n-r})
6.4 Integrals of Mean Curvature
6.4.1 Integrals of mean curvatures of hypersurfaces in E_n
6.4.2 Relations between integrals of mean curvature and quermassintegrale
6.4.3 Some particular results
6.4.4 Integrals of mean curvature of a flattened convex body
6.5 Sets of r-Planes that Intersect a Convex Set
6.5.1 Measures of the sets of r-planes that intersect a convex set
6.5.2 The integral of ... over the set etc
6.5.3 Crofton and Hadwiger's formula
6.6 Chern's Formulas
6.6.1 A density formula
6.6.2 The integral of Δ^{r+q-n}
6.6.3 Chern's formula
6.7 Santalo's Formula
6.7.1 A density formula
6.7.2 Santalo's formula
6.8 Integral of the Volume of the Intersection of Two Manifolds
6.8.1 A density formula
6.8.2 Another density formula
6.8.3 Integral of the volume σ_(r+q+n}(...)
6.9 Cbern-Yen's Kinematic Fundamental Formula
6.9.1 An important density formula
6.9.2 Chern and Yen's kinematic fundamental formula
6.9.3 Kinematic fundamental formula for convex sets
6.9.4 Integral formulas for the integrals of mean curvature
Chapter VII. Applications of Integral Geometry
7.1 Introduction to Integral Geometry in R3
7.1.1 Group of the motions in R3
7.1.2 Densities for lines and planes in R³
7 1 3 Some fundamental formlas
7.1.4 Moving cylinders
7.2 Elements of Stereology
7.2.1 Objects of study in stereology
7.2.2 General discussion
7.2.3 Intersection with random planes
7.2.4 Spherical particles
7.2.5 Nearly spherical particles
7.2.6 Intersection with random lines
7.2.7 Estimation for the number of crystals
7.3 Sufficient Conditions for One Domain to Contain Another
7.3.1 A density formula
7.3.2 A sufficient condition for one convex body to contain another
7.3.3 Sufficient conditions for one domain to contain another in R³
7.3.4 Analogues of Hadwiger's theorem in R^n (n > 3)
7.4 Kinematic Measure of a Segment of Fixed Length Within a Convex Body
7.4.1 A general formula for the kinematic measure of a segment of fixed length within a convex body
7.4.2 Transformation of formula
7.4.3 Formulas of m(l) for a cylinder
7.4.4 Kinematic measure m(l) for a right parallelepiped in R³ and Buffon problem
7.4.5 Kinematic measure m(l) for a right parallelepiped in R^n and Buffon problem
7.5 Unified Inequalities Relating to Integrals for the Power of Chords
7.5.1 Inequalities relating to integrals for the power of chords in R³
7.5.2 Applications to geometric probability
7.5.3 Inequalities of integrals for the power of chords in R^n
7.5.4 Integral geometric inequalities for moments
7.5.5 Pairs of non-intersecting random flats meeting two convex bodies
7.6 Inequalities Characterizing Simplices
7.6.1 Lemmas
7.6.2 Inequalities characterizing simplices
Index