Title and Abstract of Z. Kabluchko's course



Title: Random polytopes


Abstract:
A polytope is a convex hull of finitely many points in Euclidean space. By taking these points to be random, we obtain random polytopes. Examples include convex hulls of independent identically distributed random points (including the so-called Gaussian polytope which arises if the points have standard Gaussian distribution), convex hulls of multidimensional random walks, random projections of regular polytopes, and many others. We shall be interested in computing expectations of various functionals of such polytopes, for example the volume, the number of faces, internal and external solid angles, and some others. It turns out that there are many beautiful interrelations between these functionals. For example, Baryshnikov and Vitale observed that the number of faces has the same distribution for Gaussian polytopes as for projections of regular polytopes. The main tool used in our computations is the integral geometry of convex cones. We shall introduce the participants to this subject. In particular, we shall give various definitions of intrinsic volumes for convex cones. Also, we shall address some problems of classical geometry. For example, we shall compute the number of parts in which $n$ affine or linear hyperplanes in general position divide the $d$-dimensional space. Surpsingly, this problem is equivalent to the following one: compute the probability that the Gaussian polytope contains the origin. We shall try to stress interconnections between the conic integral geometry and various other subjects such as random matrices, the classical Sparre Andersen arcsine laws for random walks, and the high-dimensional statistics.