The geometric structure of random polytopes

During my stay at the university of Freiburg I will investigate the geometric structure of random polytopes. Random Polytopes play a central role in stochastic geometry, a part of mathematics located between geometry, analysis, and probability theory. On the one hand, the mathematical fascination of the subject led to contributions of outstanding mathematicians, on the other hand, applications like the analysis of the average complexity of algorithms and optimization have stimulated research. In the planar case there exist many results dealing with the number of vertices and the area of random polytopes. It is the aim of this project to study the analogous problem in higher dimensiones. Especially we are interested in the number of vertices and facets, as well as in the so-called intrinsic volumes, for instance, volume, and surface area, of the random polytope.