Integral geometry in spaces of constant curvature and applications to stochastic geometry

Description Phase 1

Integral geometry provides indispensable tools for the mathematical analysis of random geometric systems in Euclidean space, and conversely new developments in integral geometry are often triggered by natural tasks in stochastic geometry and statistical physics. Recent advances in translative integral geometry and tensor valuations have already been successfully applied to density functional theory in physics. We plan to explore further the integral geometry of tensor-valued functions and more general homogeneous geometric valuations in Euclidean and spherical spaces. Translative integral geometry naturally involves mixed functionals of finite sequences of convex bodies. It is an important long term goal to understand how these mixed functionals are related to mixed volumes and to express all these functionals in terms of common fundamental measures such as the flag measures. Furthermore, we will study key models of stochastic geometry including random tessellations, Boolean models or random graphs in hyperbolic space. Our previous work on Poisson hyperplanes in hyperbolic space has shown that several unexpected phenomena occur in spaces of constant negative curvature. We will develop the required tools from integral geometry in hyperbolic space needed to analyze random geometric systems in hyperbolic space and deal with specific applications for instance to random tessellations and Boolean models in non-Euclidean spaces.


Daniel Hug, Günter Last, Wolfgang Weil: Boolean models (08/2023)

Daniel Hug, Rolf Schneider: Vectorial analogues of Cauchy's surface area formula (07/2023)

Carina Betken, Daniel Hug, Christoph Thäle: Intersections of Poisson k-flats in constant curvature spaces (02/2023) published

D. Hug, R. Schneider: Poisson hyperplane tessellations (book) (01/2023)

Daniel Hug, Mario Santilli: Curvature measures and soap bubbles beyond convexity (04/2022) published

Tamara Göll, Daniel Hug: On a game of chance in Marc Elsberg's thriller "GREED" (11/2021) published

D. Hug, A Colesanti: Geometric and Functional Inequalities (CIME Summer School lecture notes) (08/2021)

Károly J. Böröczky, Daniel Hug: Reverse Alexandrov--Fenchel inequalities for zonoids (06/2021) published

Ferenc A. Bartha, Ferenc Bencs, Károly J. Böröczky, Daniel Hug: Extremizers and stability of the Betke--Weil inequality (03/2021) published

Daniel Hug, Rolf Schneider: Another look at threshold phenomena for random cones (03/2021) published

Felix Ernesti, Matti Schneider, Steffen Winter, Daniel Hug, Günter Last, Thomas Böhlke: Characterizing digital microstructures by the Minkowski-based quadratic normal tensor (06/2020) published

Daniel Hug, Rolf Schneider: Threshold phenomena for random cones (04/2020) published

Felix Herold, Daniel Hug, Christoph Thäle: Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space? (11/2019) published


  • member's portrait

    Prof. Dr. Daniel Hug

    Karlsruher Institut für Technologie
    Principal Investigator

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