Contact (Karlsruhe)
Prof. Dr. Günter Last
Karlsruhe Institute of Technology
Institut für Stochastik
Kaiserstraße 89
76133 Karlsruhe
Phone: +49-721-608 43265
Fax: +49-721-608 46691
Contact (Erlangen)
Prof. Dr. Klaus Mecke
Universität Erlangen-Nürnberg
Institut für Theoretische Physik
Staudtstraße 7
91058 Erlangen
Phone: +49-9131-85 28442
Fax: +49-9131-85 28444
Contact (Aarhus)
Department of Mathematical Sciences
Ny Munkegade 118
building 1530
8000 Aarhus C


3. Boolean models

Boolean models are a fundamental object of stochastic geometry and continuum percolation and have applications in physics, materials science and biology, for example. A Boolean model is a random closed set which is the union of compact particles of a particle process. Often it is assumed that the particle process is stationary and generated by a Poisson process, but it is of significant interest to weaken these assumptions. The goal of this project is to gain deeper insight into the geometric properties of Boolean models and the dependence on the underlying particle process by considering various functionals of Boolean models. Examples are the intrinsic volumes and Minkowski tensors as well as non-additive functionals with local extensions and flag measures. Besides mean value and density formulas this project deals with the systematical treatment of second order properties and central limit theorems. A further topic are statistical methods to estimate characteristic quantities of Boolean models such as the distribution of radii of a spherical Boolean model.

Project Members

Cooperating partners



  • Wolfgang Weil
  • Integral geometry of translation invariant functionals II: The case of general convex bodies
  • Adv. in Appl. Math. 83, 145–171 (2017)


  • Paul Goodey, Wolfram Hinderer, Daniel Hug, Jan Rataj and Wolfgang Weil
  • A flag representation of projection functions
  • Adv. Geom. (to appear) , (2016)
  • Julia Hörrmann and Anne Marie Svane
  • Local digital algorithms applied to Boolean models
  • Scand. J. Stat. (to appear) (2016)
  • Julia Hörrmann and Wolfgang Weil
  • Valuations and Boolean models
  • to appear in: Tensor Valuations and their Applications in Stochastic Geometry and Imaging (edited by Markus Kiderlen and Eva B. Vedel Jensen), Lecture Notes in Mathematics (2016)
  • Daniel Hug and Matthias Reitzner
  • Introduction to Stochastic Geometry
  • to appear in: Stochastic analysis for Poisson point processes: Malliavin calculus, Wiener-Ito chaos expansions and stochastic geometry (edited by Giovanni Peccati and Matthias Reitzner, Springer & Bocconi Series (2016)) (2016)
  • Daniel Hug, Michael A. Klatt, Günter Last and Matthias Schulte
  • Second order analysis of geometric functionals of Boolean models
  • to appear in: Tensor Valuations and their Applications in Stochastic Geometry and Imaging (edited by Markus Kiderlen and Eva B. Vedel Jensen), Lecture Notes in Mathematics (2016)
  • Daniel Hug, Günter Last and Matthias Schulte
  • Second-order properties and central limit theorems for geometric functionals of Boolean models
  • Ann. Appl. Probab. 26(1), 73–135 (2016)
  • Klatt, Michael A.
  • Morphometry of random spatial structures in physics
  • FAU University Press, Erlangen 2016
    note: PhD thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg


  • Günter Last and Hermann Thorisson
  • Construction and characterization of stationary and mass-stationary random measures on R^d
  • Stochastic Process. Appl. 125(12), 4473–4488 (2015)
  • Scholz, Christian and Wirner, Frank and Klatt, Michael A. and Hirneise, Daniel and Schröder-Turk, Gerd E. and Mecke, Klaus and Bechinger, Clemens
  • Direct relations between morphology and transport in Boolean models
  • Phys. Rev. E 92(4), 043023 (2015)
  • Wolfgang Weil
  • Integral geometry of translation invariant functionals I: The polytopal case
  • Adv. in Appl. Math. 66, 46–79 (2015)


  • Julia Hörrmann, Daniel Hug, Michael Klatt, and Klaus Mecke
  • Minkowski tensor density formulas for Boolean models
  • Adv. in Appl. Math. 55, 48–85 (2014)
  • Daniel Hug, Günter Last, Zbynek Pawlas, and Wolfgang Weil
  • Statistics for Poisson models of overlapping spheres
  • Adv. in Appl. Probab. 46, 937–962 (2014)
  • Svane, Anne Marie
  • Local digital estimators of intrinsic volumes for Boolean models and in the design based setting
  • Adv. in Appl. Probab. 46, 35–58 (2014)


  • Daniel Hug, Jan Rataj, and Wolfgang Weil
  • A product integral representation of mixed volumes of two convex bodies
  • Adv. Geom. 13, 633–662 (2013)
  • Daniel Hug, Ines Türk, and Wolfgang Weil
  • Flag measures for convex bodies
  • Fields Institute Communications (eds. Monika Ludwig, Vitali D. Milman, Vladimir Oestov, Nicole Tomczak-Jaegermann) 68, 145–187 (2013)
  • Matthias Reitzner, Matthias Schulte, and Christoph Thäle
  • Limit theory for the Gilbert graph
  • Preprint (2013)


  • Günter Last and Ryszard Szekli
  • Comparisons and asymptotics for empty space hazard functions of germ-grain models
  • Adv. in Appl. Probab. 43, 942–962 (2011)