Now available in the Cambridge Mathematical Library, the classic work from Luis Santalo. Integral geometry originated with problems on geometrical probability and convex bodies. Its later developments, however, have proved useful in several fields ranging from pure mathematics (measure theory, continuous groups) to technical and applied disciplines (pattern recognition, stereology). The book is a systematic exposition of the theory and a compilation of the main results in the field. The volume can be used to complement courses on differential geometry, Lie groups or probability, or differential geometry. It is ideal both as a reference and for those wishing to enter the field.

Editor's Statement

Foreword

Preface

Part I: INTEGRAL GEOMETRY IN THE PLANE

Chapter 1. Convex Sets in the Plane

 1. Introduction

 2. Envelope of a Family of Lines

 3. Mixed Areas of Minkowski

 4. Some Special Convex Sets

 5. Surface Area of the Unit Sphere and Volume of the Unit Ball

 6. Notes and Exercises

Chapter 2. Sets of Points and Poisson Processes in the Plane

 1. Density for Sets of Points

 2. First Integral Formulas

 3. Sets of Triples of Points

 4. Homogeneous Planar Poisson Point Processes

 5. Notes

Chapter 3. Sets of Lines in the Plane

 1. Density for Sets of Lines

 2. Lines That Intersect a Convex Set or a Curve

 3. Lines That Cut or Separate Two Convex Sets

 4. Geometric Applications

 5. Notes and Exercises

Chapter 4. Pairs of Points and Pairs of Lines

 1. Density for Pairs of Points

 2. Integrals for the Power of the Chords of a Convex Set

 3. Density for Pairs of Lines

 4. Division of the Plane by Random Lines

 5. Notes

Chapter 5. Sets of Strips in the Plane

 1. Density for Sets of Strips

 2. Buffon's Needle Problem

 3. Sets of Points, Lines, and Strips

 4. Some Mean Values

 5. Notes

Chapter 6. The Group of Motions in the Plane: Kinematic Density

 1. The Group of Motions in the Plane

 2. The Differential Forms on m

 3. The Kinematic Density

 4. Sets of Segments

 5. Convex Sets That Intersect Another Convex Set

 6. Some Integral Formulas

 7. A Mean Value; Coverage Problems

 8. Notes and Exercises

Chapter 7. Fundamental Formulas of Poincare and Blaschke

 1. A New Expression for the Kinematic Density

 2. Poincare's Formula

 3. Total Curvature of a Closed Curve and of a Plane Domain

 4. Fundamental Formula of Blaschke

 5. The lsoperimetric Inequality

 6. Hadwiger's Conditions for a Domain to Be Able to Contain Another

 7. Notes

Chapter 8. Lattices of Figures

 1. Definitions and Fundamental Formula

 2. Lattices of Domains

 3. Lattices of Curves

 4. Lattices Of Points

 5. Notes and Exercise

Part I1. GENERAL INTEGRAL GEOMETRY

Chapter 9. Differential Forms and Lie Groups

 1. Differential Forms

 2. Pfaffian Differential Systems

 3. Mappings of Differentiable Manifolds

 4. Lie Groups; Left and Right Translations

 5. Left-lnvariant Differential Forms

 6. Maurer-Cartan Equations

 7. lnvariant Volume Elements of a Group: Unimodular Groups

 8. Notes and Exercises

Chapter 10. Density and Measure in Homogeneous Spaces

 1. Introduction

 2. lnvariant Subgroups and Quotient Groups

 3. Other Conditions for the Existence of a Density on Homo-geneous Spaces

 4. Examples

 5. Lie Transformation Groups

 6. Notes and Exercises

Chapter 11. The Affine Groups

 1. The Groups of Affine Transformations

 2. Densities for Linear Spaces with Respect to Special Homo-geneous Affinities

 3. Densities for Linear Subspaces with Respect to the Special Nonhomogeneous Affine Group

 4. Notes and Exercises

Chapter 12. The Group of Motions in En

 1. Introduction

 2. Densities for Linear Spaces in En

 3. A Differential Formula

 4. Density for r-Planes about a Fixed q-Plane

 5. Another Form of the Density for r-Planes in En

 6. Sets of Pairs of Linear Spaces

 7. Notes

Part III INTEGRAL GEOMETRY IN En

Chapter 13. Convex Sets in En

 1. Convex Sets and Quermassintegrale

 2. Cauchy's Formula

 3. Parallel Convex Sets; Steiner's Formula

 4. Integral Formulas Relating to the Projections of a Convex Set on Linear Subspaces

 5. Integrals of Mean Curvature

 6. Integrals of Mean Curvature and Quermassintegrale

 7. Integrals of Mean Curvature of a Flattened Convex Body

 8. Notes

Chapter 14. Linear Subspaces, Convex Sets, and Compact Manifolds

 1. Sets of r-Planes That Intersect a Convex Set

 2. Geometric Probabilities

 3. Crofton's Formulas in En

 4. Some Relations between Densities of Linear Subspaces

 5. Linear Subspaces That Intersect a Manifold

 6. Hypersurfaces and Linear Spaces

 7. Notes

Chapter 15. The Kinematic Density in En

 1. Formulas on Densities

 2. Integral of the Volume σr+q-n(Mr∩Mq)

 3. A Differential Formula

 4. The Kinematic Fundamental Formula

 5. Fundamental Formula for Convex Sets

 6. Mean Values for the Integrals of Mean Curvature

 7. Fundamental Formula for Cylinders

 8. Some Mean Values

 9. Lattices in En.

 10. Notes and Exercise

Chapter 16. Geometric and Statistical Applications; Stereology

 1. Size Distribution of Particles Derived from the Size Distribution of Their Sections

 2. Intersection with Random Planes

 3. Intersection with Random Lines

 4. Notes

Part IV. INTEGRAL GEOMETRY IN SPACES OF CONSTANT CURVATURE

Chapter 17. Noneuclidean Integral Geometry

 1. The n-Dimensional Noneuclidean Space

 2. The Gauss-Bonnet Formula for Noneuclidean Spaces

 3. Kinematic Density and Density for r-Planes

 4. Sets of r-Planes That Meet a Fixed Body

 5. Notes

Chapter 18. Crofton's Formulas and the Kinematic Fundamental Formula in Noneuclidean Spaces

 1. Crofton's Formulas

 2. Dual Formulas in Elliptic Space

 3. The Kinematic Fundamental Formula in Noneuclidean Spaces

 4. Steiner's Formula in Noneuclidean Spaces

 5. An Integral Formula for Convex Bodies in Elliptic Space

 6. Notes

Chapter 19. Integral Geometry and Foliated Spaces; Trends in Integral Geometry

 1. Foliated Spaces

 2. Sets of Geodesics in a Riemann Manifold

 3. Measure of Two-Dimensional Sets of Geodesics

 4. Measure of(2n - 2)-Dimensional Sets of Geodesics

 5. Sets of Geodesic Segments

 6. Integral Geometry on Complex Spaces

 7. Symplectic Integral Geometry

 8. The Integral Geometry of Gelfand

 9. Notes

Appendix. Differential Forms and Exterior Calculus

 1. Differential Forms and Exterior Product

 2. Two Applications of the Exterior Product

 3. Exterior Differentiation

 4. Stokes' Formula

 5. Comparison with Vector Calculus in Euclidean Three-Dimensional Space

 6. Differential Forms over Manifolds

Bibliography and References

Author Index

Subject index