## Stochastic Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 13-18, 2004Stochastic Geometry is the mathematical discipline which studies mathematical models for random geometric structures, as they appear frequently in almost all natural sciences or technical fields. Although its roots can be traced back to the 18th century (the Buffon needle problem), the modern theory of random sets was founded by D. Kendall and G. Matheron in the early 1970's. Its rapid development was influenced by applications in Spatial Statistics and by its close connections to Integral Geometry. The volume "Stochastic Geometry" contains the lectures given at the CIME summer school in Martina Franca in September 2004. The four main lecturers covered the areas of Spatial Statistics, Random Points, Integral Geometry and Random Sets, they are complemented by two additional contributions on Random Mosaics and Crystallization Processes. The book presents an up-to-date description of important parts of Stochastic Geometry. |

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### Contents

1 | |

Moments and Summary Statistics | 26 |

Conditioning | 42 |

Modelling and Statistical Inference | 57 |

References | 73 |

Minimal Caps and a General Result | 80 |

Proofs of the Properties of the Mregions | 87 |

Proof of Theorem 3 1 | 95 |

Translative Integral Geometry | 151 |

Measures on Spaces of Flats | 164 |

References | 181 |

Mean Values of Additive Functionals | 198 |

Contact Distributions | 231 |

References | 243 |

Voronoi and Delaunay Mosaics | 253 |

Kendalls Conjecture | 260 |

Expectation of fkKn | 101 |

Lattice Polytopes | 108 |

Segments on the Surface of K | 114 |

From Hitting Probabilities to Kinematic Formulae | 120 |

Localizations and Extensions | 136 |

On the Evolution Equations of Mean Geometric Densities | 267 |

The Evolution Equation of Mean Densities for the Stochastic | 274 |

References | 280 |