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



1. Tensor valuations

Tensor valuations are functionals on convex or more general sets which are additive (valuations) and take their values in the space of symmetric tensors. They satisfy integral-geometric formulas and can be used to describe the morphology of spatial structure. In this project, we consider new kinematic formulas of additive, intersectional, rotational and Crofton type for tensor valuations and general classification results for tensor-valued measures. Algorithms are implemented to numerically calculate tensorial valuations for a wide class of data. The Minkowski tensors are used to investigate the complex spatial structures of porous materials, granular packings and fluids. A density functional theory of aspherical particle fluids based on Minkowski tensors is developed, and the morphometric approach for thermodynamic properties of confinded fluids is explored. Tensor functionals play an important role as geometric descriptors in all projects of the research unit.

Project Members

Cooperating partners



  • Andreas Bernig and Daniel Hug
  • Kinematic formulas for tensor valuations
  • J. Reine Angew. Mathematik (to appear) (2016)
  • Andreas Bernig and Daniel Hug
  • Integral geometry and algebraic structures for tensor valuations
  • Preprint (2016)
  • Paul Goodey, Daniel Hug, and Wolfgang Weil
  • Kinematic formulas for area measures
  • Indiana Univ. Math. J. (to appear) (2016)
  • Julia Hörrmann and Astrid Kousholt
  • Reconstruction of convex bodies from moments
  • Preprint (2016)
  • Daniel Hug and Rolf Schneider
  • Rotation covariant local tensor valuations on convex bodies.
  • Communications in Contemporary Mathematics (to appear) (2016)
  • Daniel Hug and Jan A. Weis
  • Crofton formulae for tensor-valued curvature measures
  • Preprint (2016)
  • Daniel Hug and Jan Rataj
  • Mixed curvature measures of translative integral geometry
  • Preprint (2016)
  • Daniel Hug and Rolf Schneider
  • SO(n) covariant local tensor valuations on polytopes.
  • Preprint (2016)
  • Daniel Hug and Rolf Schneider
  • Tensor valuations and their local versions
  • 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)
  • Eva B. Vedel Jensen and Markus Kiderlen (eds.)
  • Tensor Valuations and their Applications in Stochastic Geometry and Imaging
  • To appear as Lecture Notes in Mathematics. (2016)
  • Eva B. Vedel Jensen and Markus Kiderlen
  • Rotation invariant valuations
  • 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)
  • Michael A. Klatt, Günter Last, Klaus Mecke, Claudia Redenbach, Fabian M. Schaller and Gerd E.Schröder-Turk
  • Cell shape analysis of random tessellations based on Minkowski tensors
  • arXiv:1606.07653 [cond-mat] (2016)
    note: arXiv: 1606.07653
  • Svane, Anne Marie and Jensen, Eva B. Vedel
  • Rotational Crofton formulae for Minkowski tensors and some affine counter parts.
  • Preprint, CSGB Research Reports 2016-13. (2016)
  • Steffen Winter
  • Localization results for Minkowski contents
  • Preprint (2016)


  • Wolfram Hinderer and Daniel Hug and Wolfgang Weil
  • Extensions of translation invariant valuations on polytopes
  • Mathematika 61, 236–258 (2015)
  • Daniel Hug and Rolf Schneider
  • Hölder continuity of normal cycles and of support measures of convex bodies
  • Arch. Math. 104, 83–92 (2015)
  • Astrid Kousholt, Markus Kiderlen, and Daniel Hug
  • Surface tensor estimation from linear sections
  • Math. Nachr. 288, 1647–1672 (2015)
  • Philipp Schönhöfer and Klaus Mecke
  • The Shape of Anisotropic Fractals: Scaling of Minkowski Functionals
  • pages 39-52 in: Progress in Probability, Vol. 70: Fractal Geometry and Stochastics V (editor(s): Christoph Bandt, Kenneth Falconer and Martina Zähle), Springer International Publishing, 2015
  • Steffen Winter
  • Minkowski content and fractal curvatures of self-similar tilings and generator formulas for self-similar sets
  • Adv. Math. 274, 285–322 (2015)


  • Myfanwy E. Evans and Roland Roth
  • Shaping the skin: the interplay of mesoscale geometry and corneocyte swelling
  • Physical Review Letters 112(3), 038102:1–5 (2014)
  • Myfanwy E. Evans and Roland Roth
  • Solvation of a sponge-like geometry
  • Pure and Applied Chemistry 86(2), 173–179 (2014)
  • Daniel Hug and Rolf Schneider
  • Local tensor valuations
  • Geom. Funct. Anal. 24, 1516–1564 (2014)
  • Eva B. Vedel Jensen and Johanna F. Ziegel
  • Local stereology of tensors of convex bodies
  • Methodol. Comput. Appl. Probab. 16, 263–282 (2014)
  • Dusan Pokorny and Steffen Winter
  • Scaling exponents of curvature measures
  • J. Fractal Geometry 1, 177-219 (2014)
  • Ólöf Thórisdóttir and Markus Kiderlen
  • The invariator principle in convex geometry
  • Adv. in Appl. Math. 58, 63–87 (2014)


  • Jérémy Auneau-Cognacq, Johanna Ziegel, and Eva B. Vedel Jensen
  • Rotational integral geometry of tensor valuations
  • Adv. in Appl. Math. 50, 429–444 (2013)
  • Daniel Hug, Jan Rataj, and Wolfgang Weil
  • A product integral representation of mixed volumes of two convex bodies
  • Adv. Geom. 13, 633–662 (2013)
  • Michel L. Lapidus, Erin Pearse, and Steffen Winter
  • Minkowski measurability results for self-similar tilings and fractals with monophase generators
  • pages 185-203 in: Contemp. Math 600: Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I: Fractals in Pure Mathematics (editor(s): Michel L. Lapidus, Erin Pearse, and Machiel van Frankenhuijsen), 2013
  • Jan Rataj and Steffen Winter
  • Characterization of Minkowski measurability in terms of surface area
  • J. Math. Anal. Appl. 400, 120–132 (2013)
  • Gerd E. Schröder-Turk, Walter Mickel, Sebastian C. Kapfer, Fabian M. Schaller, Boris Breidenbach, Daniel Hug, and Klaus Mecke
  • Minkowski tensors of anisotropic spatial structure
  • New J. Phys. 15, 083028 (2013)
  • Steffen Winter and Martina Zähle
  • Fractal curvature measures of self-similar sets
  • Adv. Geom. 13, 229-244 (2013)


  • Sebastian C. Kapfer, Walter Mickel, Klaus Mecke, and Gerd E. Schröder-Turk
  • Jammed spheres: Minkowski tensors reveal onset of local crystallinity
  • Physical Review E 85(3), 030301:1–4 (2012)
  • Walter Mickel, Gerd E. Schröder-Turk, and Klaus Mecke
  • Tensorial Minkowski functionals of triply periodic minimal surfaces
  • Interface Focus 2(5), 623–633 (2012)
  • Mohammad Saadatfar, Manas Mukherjee, Mahyar Madadi, Gerd E. Schröder-Turk, Francisco Garcia-Moreno, Fabian M. Schaller, Stefan Hutzler, Adrian P. Sheppard, John Banhart, and Upadrasta Ramamurty
  • Structure and deformation correlation of closed-cell aluminium foam subject to uniaxial compression
  • Acta Materialia 60(8), 3604–3615 (2012)
  • Gerd E. Schröder-Turk, Walter Mickel, Sebastian C. Kapfer, Fabian M. Schaller, Boris Breidenbach, Daniel Hug, and Klaus Mecke
  • On the volume of the zero cell of a class of isotropic Poisson hyperplane tessellations
  • Preprint, p. (2012)


  • Roland Roth, Klaus Mecke, and Martin Oettel
  • Fundamental measure theory for hard disks: fluid and solid
  • The Journal of Chemical Physics 136(8), 081101 (2011)
  • Gerd E. Schröder-Turk, Walter Mickel, Sebastian C. Kapfer, Michael A. Klatt, Fabian M. Schaller, Matthias J. F. Hoffmann, Nicola Kleppmann, Patrick Armstrong, Alexandra Inayat, Daniel Hug, Martin Reichelsdorfer, Wolfgang Peukert, Wilhelm Schwieger, and Klaus Mecke
  • Minkowski Tensor Shape Analysis of Cellular, Granular and Porous Structures
  • Advanced Materials 23(22-23), 2535–2553 (2011)