Weak convergence of random geometric objects in non-Euclidean spaces

The analysis of conventional convex hulls of random points sampled from the uniform distribution on a compact convex subset of the d-dimensional Euclidean space is the classical topic in stochastic geometry. In the past years there has been a splash of activity around various generalized concepts of convexity both in geometric and probabilistic communities. The talk is devoted to the discussion of two particular models of this kind. In the first model we consider a sample picked from the uniform distribution on the upper semi-sphere and analyze its spherical convex hull. In the second model the sample is taken from the uniform distribution in a convex body K in the d-dimensional Euclidean space and we focus on the analysis of its K-convex hull, that is, intersection of all affine translates of K which contain the sample. Considered from an appropriate viewpoint, these two models incorporating generalized notion of convexity exhibit a similar behavior which turns out to be very different from such in the classical setting.