Skip to Main content Skip to Navigation
Habilitation à diriger des recherches

Mosaïques, enveloppes convexes et modèle Booléen : quelques propriétés et rapprochements

Abstract : This dissertation is devoted to three classical models of random geometry: tessellations, convex hulls and Boolean model. The first section deals with isotropic Poisson hyperplane tessellations. We study zero-cells which are random convex polytopes of the Euclidean space. Particular cases include the typical Poisson-Voronoi cell and the Crofton cell. We provide an explicit formula for the distribution of the number of sides in dimension two. We investigate its asymptotic behaviour and connection is made with Sylvester's problem for points in convex position. We then investigate distributional properties of the circumscribed radius and limit theorems are proved for describing the asymptotic behaviour of the zero-cell with large inradius. This (and also our use of fundamental frequency) allows us to specify in some cases the statement of D. G. Kendall's conjecture. In the second section, we are interested in convex hulls of isotropic Poisson point processes in the unit-ball. A large deviation type result is shown for the number of vertices. We then prove the convergence of the rescaled boundary of the polytope and we deduce from it extreme value results, variance estimates, central limit theorems and invariance principles for some characteristics. The third section is about Boolean type covering models of the Euclidean space. In a first work we model a multicracking process with a covering model without overlappings. Next we study the convergence of the connected component of the origin of a Boolean model to the Crofton cell in dimension two. Finally we investigate the visibility function of this connected component and we prove precise distributional estimates and extreme value results.
Document type :
Habilitation à diriger des recherches
Complete list of metadatas

Cited literature [119 references]  Display  Hide  Download
Contributor : Pierre Calka <>
Submitted on : Monday, January 18, 2010 - 3:31:12 PM
Last modification on : Friday, April 10, 2020 - 5:20:14 PM
Long-term archiving on: : Friday, June 18, 2010 - 1:04:56 AM


  • HAL Id : tel-00448249, version 1


Pierre Calka. Mosaïques, enveloppes convexes et modèle Booléen : quelques propriétés et rapprochements. Mathématiques [math]. Université René Descartes - Paris V, 2009. ⟨tel-00448249⟩



Record views


Files downloads