Contact (Karlsruhe)
Prof. Dr. Günter Last
Karlsruhe Institute of Technology
Institut für Stochastik
Kaiserstraße 89
76133 Karlsruhe
Germany
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
Germany
Phone: +49-9131-85 28442
Fax: +49-9131-85 28444
Contact (Aarhus)
CSGB
Department of Mathematical Sciences
Ny Munkegade 118
building 1530
8000 Aarhus C
Denmark

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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

Publications

2018

  • Hug, Daniel and Rataj, Jan
  • Mixed curvature measures of translative integral geometry
  • Geometriae Dedicata 195(1), 101–120 (2018)
  • Steffen Winter
  • Localization results for Minkowski contents
  • J. Lond. Math. Soc. (to appear) (2018)
  • Hug, Daniel and Thaele, Christoph
  • Splitting tessellations in spherical spaces
  • Preprint (2018)
    note: arXiv: 1804.08740
  • Hug, Daniel and Weis, Jan A.
  • Kinematic formulae for tensorial curvature measures
  • Annali di Matematica Pura ed Applicata (1923 -) 197(5), 1349–1384 (2018)
  • Hug, Daniel and Rataj, Jan and Weil, Wolfgang
  • Flag representations of mixed volumes and mixed functionals of convex bodies
  • J. Math. Anal. Appl. 460(2) (2018)
  • Schneider, Rolf
  • Conic support measures
  • Preprint (2018)
    note: arXiv: 1807.03614
  • Schneider, R.
  • Intersection probabilities and kinematic formulas for polyhedral cones
  • Acta Math. Hungar. 155(1), 3–24 (2018)
  • Schneider, Rolf
  • Polyhedral Gauss–Bonnet theorems and valuations
  • Beitr. Algebra Geom. 59(2), 199–210 (2018)
  • Simon Weis and Gerd E Schröder-Turk and Matthias Schröter
  • Structural similarity between dry and wet sphere packings
  • Preprint (2018)
    note: arXiv: 1808.04342
  • Matthias C. Bott and Felix Winterhalter and Matthieu Marechal and Abhinav Sharma and Joseph M. Brader and René Wittmann
  • Isotropic-nematic transition of self-propelled rods in three dimensions
  • Phys. Rev. E 98, 012601 (2018)
  • Philipp W. A. Schönhöfer and Gerd E. Schröder-Turk and Matthieu Marechal
  • Density functional theory for hard uniaxial particles: Complex ordering of pear-shaped and spheroidal particles near a substrate
  • J. Chem. Phys. 148, 124104 (2018)

2017

  • Bernig, Andreas and Hug, Daniel
  • Integral geometry and algebraic structures for tensor valuations
  • pages 79–109 in: Lecture Notes in Math., Vol. 2177: Tensor valuations and their applications in stochastic geometry and imaging (editor(s): Jensen, Eva B. Vedel and Kiderlen, Markus), Springer, Cham, 2017
  • Paul Goodey, Daniel Hug, and Wolfgang Weil
  • Kinematic formulas for area measures
  • Indiana Univ. Math. J. 66, 997–1018 (2017)
  • Goodey, Paul and Hinderer, Wolfram and Hug, Daniel and Rataj, Jan and Weil, Wolfgang
  • A flag representation of projection functions
  • Adv. Geom. 17(3), 303–322 (2017)
  • Hug, Daniel and Schneider, Rolf
  • Rotation covariant local tensor valuations on convex bodies
  • Communications in Contemporary Mathematics 19(05), 1650061-1–31 (2017)
  • Hug, Daniel and Weis, Jan A.
  • Crofton formulae for tensor-valued curvature measures
  • pages 111–156 in: Lecture Notes in Math., Vol. 2177: Tensor valuations and their applications in stochastic geometry and imaging (editor(s): Jensen, Eva B. Vedel and Kiderlen, Markus), Springer, Cham, 2017
  • Hug, Daniel and Schneider, Rolf
  • SO(n) Covariant local tensor valuations on polytopes
  • The Michigan Mathematical Journal 66(3), 637–659 (2017)
  • Hug, Daniel and Kiderlen, Markus and Svane, Anne Marie
  • Voronoi-Based Estimation of Minkowski Tensors from Finite Point Samples
  • Discrete Comput. Geom. 57(3), 545–570 (2017)
  • Hug, Daniel and Schneider, Rolf
  • Tensor valuations and their local versions
  • pages 27–65 in: Lecture Notes in Math., Vol. 2177: Tensor valuations and their applications in stochastic geometry and imaging (editor(s): Jensen, Eva B. Vedel and Kiderlen, Markus), Springer, Cham, 2017
  • Jensen, Eva B. Vedel and Kiderlen, Markus (eds.)
  • Tensor valuations and their applications in stochastic geometry and imaging
  • Lecture Notes in Mathematics, Vol. 2177: Springer, Cham, 2017
  • Eva B. Vedel Jensen and Markus Kiderlen
  • Rotation invariant valuations.
  • pages 185–212 in: Tensor valuations and their applications in stochastic geometry and imaging. Based on the presentations at the workshop, Sandbjerg Manor, Denmark, September 21–26, 2014, Cham: Springer, 2017
  • Weis, Jan Andreas
  • Tensorial Curvature Measures in Integral Geometry
  • PhD thesis, Karlsruhe Institute of Technology (KIT), Karlsruhe 2017
  • Klatt, Michael A. and Last, Günter and Mecke, Klaus and Redenbach, Claudia and Schaller, Fabian M. and Schröder-Turk, Gerd E.
  • Cell Shape Analysis of Random Tessellations Based on Minkowski Tensors
  • pages 385–421 in: Lecture Notes in Mathematics, Vol. 2177: Tensor Valuations and Their Applications in Stochastic Geometry and Imaging (editor(s): Vedel Jensen, Eva B. and Kiderlen, Markus), Springer International Publishing, Cham 2017
  • Klatt, Michael A. and Schröder-Turk, Gerd E. and Mecke, Klaus
  • Mean-intercept anisotropy analysis of porous media. I. Analytic formulae for anisotropic Boolean models
  • Med. Phys. 44(7), 3650–3662 (2017)
  • Klatt, Michael A. and Schröder-Turk, Gerd E. and Mecke, Klaus
  • Mean-intercept anisotropy analysis of porous media. II. Conceptual shortcomings of the MIL tensor definition and Minkowski tensors as an alternative
  • Med. Phys. 44(7), 3663–3675 (2017)
  • Svane, Anne Marie and Vedel Jensen, Eva B.
  • Rotational Crofton formulae for Minkowski tensors and some affine counterparts
  • Adv. in Appl. Math. 91, 44–75 (2017)
  • Wolfgang Weil
  • Integral geometry of translation invariant functionals II: The case of general convex bodies
  • Adv. in Appl. Math. 83, 145–171 (2017)
  • Hug, Daniel and Weis, Jan A.
  • Integral geometric formulae for Minkowski tensors
  • Preprint (2017)
    note: arXiv: 1712.09699
  • Hug, Daniel and Weil, Wolfgang
  • Determination of Boolean models by mean values of mixed volumes
  • Preprint (2017)
    note: arXiv: 1712.08241
  • Hug, Daniel and Weis, Jan A.
  • Crofton Formulae for Tensorial Curvature Measures: The General Case
  • pages 39–60 in: Analytic Aspects of Convexity (editor(s): Bianchi, Gabriele and Colesanti, Andrea and Gronchi, Paolo), Springer International Publishing, Cham 2017
  • Colesanti, Andrea and Hug, Daniel and Saor\'in Gómez, Eugenia
  • Monotonicity and concavity of integral functionals involving area measures of convex bodies
  • Communications in Contemporary Mathematics 19(02), 1650033-1–26 (2017)
  • Schneider, R.
  • Combinatorial identities for polyhedral cones
  • Algebra i Analiz, p. 279–295 (2017)
    note: translation in St. Petersburg Math. J. \textbf29 (2018), no. 1, 209–221
  • Schneider, Rolf
  • Valuations on Convex Bodies: The Classical Basic Facts
  • pages 1–25 in: Tensor Valuations and Their Applications in Stochastic Geometry and Imaging (editor(s): Jensen, Eva B. Vedel and Kiderlen, Markus), Springer International Publishing, Cham 2017
  • Astrid Kousholt
  • Reconstruction of $n$-dimensional convex bodies from surface tensors.
  • Adv. Appl. Math. 83, 115–144 (2017)

2016

  • Julia Schulte and Astrid Kousholt
  • Reconstruction of convex bodies from moments
  • Discrete and Computational Geometry (accepted with minor revision), arXiv:1605.06362 (2016)

2015

  • Andreas Bernig and Daniel Hug
  • Kinematic formulas for tensor valuations
  • J. Reine Angew. Math. 2018(736), 141–191 (2015)
  • 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
  • Wolfgang Weil
  • Integral geometry of translation invariant functionals I: The polytopal case
  • Adv. in Appl. Math. 66, 46–79 (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)

2014

  • 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)
  • Julia Hörrmann
  • The method of densities for non-isotropic Boolean models
  • PhD thesis, Karlsruhe Institute of Technology (KIT), Karlsruhe 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)
  • Colesanti, Andrea and Hug, Daniel and Saor\'in Gómez, Eugenia
  • A Characterization of Some Mixed Volumes via the Brunn–Minkowski Inequality
  • J. Geom. Analysis 24(2), 1064–1091 (2014)
  • Goodey, Paul and Weil, Wolfgang
  • Sums of sections, surface area measures, and the general Minkowski problem
  • J. Differential Geom. 97(3), 477–514 (2014)
  • M. Marechal and S. Korden and K. Mecke
  • Deriving fundamental measure theory from the virial series: Consistency with the zero-dimensional limit
  • Phys. Rev. E 90, 042131 (2014)

2013

  • 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)
  • G E Schröder-Turk and W Mickel and S C Kapfer and F M Schaller and B Breidenbach and D Hug and K Mecke
  • Minkowski tensors of anisotropic spatial structure
  • New Journal of Physics 15(8), 083028 (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)
  • Schneider, Rolf
  • Local tensor valuations on convex polytopes
  • Monatsh. Math. 171(3–4), 459–479 (2013)
  • Markus Kiderlen
  • Introduction into integral geometry and stereology.
  • pages 21–48 in: Stochastic geometry, spatial statistics and random fields. Asymptotic methods. Selected papers based on the presentations at the summer academy on stochastic geometry, spatial statistics and random fields, Söllerhaus, Germany, September 13–26, 2009, Berlin: Springer, 2013

2012

  • 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)

2011

  • 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)