Convex and Stochastic Geometry - Einzelansicht

Die Vorlesung richtet sich an Masterstudenten, die im Bachelorstudium die Vorlesungen Stochastik und  Wahrscheinlichkeitstheorie besucht haben. Sie kann als ein Bestandteil sowohl des Moduls "Wahrscheinlichkeitstheorie und ihre Anwendungen" als auch des Moduls "Stochastische Prozesse" gewählt werden.

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The preknowledge in Measure Theory are highly recommended, the preknowledge in Probability Theory are necessary for the second part.

Bearbeiten von Übungen und Bestehen einer Klausur bzw. mündlichen Prüfung.

In this course we will study some basic concepts of Convex and Integral Geometry and at the end we will get in touch with a few classical results from Stochastic geometry. In the first part of the lecture course we will construct a metric space of compact subsets and consider special functionals of convex sets, called valuations. An important example of such functionals are intrinsic volumes, which we will study in details. In the second part of this lecture course we will work with the random convex sets.

As an example we consider the following model: for a given set of random points consider a smallest convex set, containing all of them (the so-called convex hull). This convex hull give rise to the random polytope, for which there exists a number problems both classical and open by now. In this course we in particular prove the result of Wendel, which gives an exact formula for the probability that a random polytope contains zero. Assuming additionally that these random points are distributed uniformly inside some convex set of volume one, another typical question would be: For which convex set is the expected volume of the random polytope minimal/maximal?