## Integral Geometry and Geometric ProbabilityIntegral geometry originated with problems on geometrical probability and convex bodies. Its later developments have proved to be 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. |

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### Contents

INTEGRAL GEOMETRY IN THE PLANE | 1 |

Part II GENERAL INTEGRAL GEOMETRY | 143 |

Part III INTEGRAL GEOMETRY IN En | 215 |

Part IV INTEGRAL GEOMETRY IN SPACES OF CONSTANT CURVATURE | 299 |

Bibliography and References | 363 |

395 | |

399 | |

### Common terms and phrases

1-forms affine angle applied area element area F Assume boundary bounded convex set breadth Chapter chord congruent consider contained convex body convex set convex set K0 coordinates cylinder defined differentiable manifold differential form dimension distance distribution domain equal euclidean space exterior product fixed fundamental formula fundamental regions geodesics geometric probability given group of motions Hence homogeneous space hyperplanes integral formulas integral geometry integral of mean intersection points invariant density isoperimetric inequality kinematic density L. A. Santalo lattice left-invariant Lie group line G line segment linear linear subspaces M. I. Stoka Math matrix mean curvature mean number mean value measure n-dimensional n.e. space number of intersection orthogonal parallel particles perimeter perpendicular plane Poisson polygons problem r-planes random lines Section segment of length sets K1 strips subgroup tangent theorem transformation triangle unimodular unit sphere vector