Curvature Measures and Integral Geometry

• Applicants: Professor Dr. Daniel Hug; Professor Dr. Wolfgang Weil (†); Privatdozent Dr. Steffen Winter
• Subject Area: Mathematics
• Term: from 2010 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 160223306

In the current project, the investigation of curvature measures in euclidean spaces has been advanced in two directions. In a first subproject, curvature related quantities of very general sets have been studied via the approximation with parallel sets and the asymptotic behaviour of such approximations has been explored. Here, very general connections between the asymptotic behaviour of the volume and the surface area of the parallel sets have been found, and this has led to a good understanding of the global aspects of this approximation. Moreover, the existence of fractal curvature measures of self-similar sets has been established in a general framework. In the second subproject, extensions of curvature measures of convex bodies to measures on flag manifolds have been treated. The properties of such flag measures have been studied and it has been discussed how flag measures can be used for classical integral formulas in convex geometry and for the investigation of valuations. Many of the problems that have been addressed are not completely resolved so far. This should be done in the second phase of the project. In particular, we plan to localize, in a measure theoretic setting, the results obtained in the first subproject concerning the connection between volume and surface area of general sets. In addition, we shall study especially those self-similar sets whose curvature measures do not scale with the dimension, with the aim of understanding the geometric meaning of the scaling exponents. In the context of flag measures, further integral representations of mixed volumes and of special functionals (such as the support function) should be developed. Another goal is to find a description of homogeneous valuations by means of flag measures. Finally, applications to stochastic geometry, for instance to Boolean models, still remain to be considered. These applications require the investigation of injectivity properties of operators which are associated with flag measures.

• DFG Programme: Research Grants