# Random polytopes and phenomena in high dimensions

Random polytopes in $\mathbb{R}^n$ arise by taking the convex hull of a finite number of random points that are distributed according to some fixed probability distribution (for example, the uniform distribution on a convex body or the standard Gaussian distribution). Alternatively, random polytopes can also be generated as intersection of random half-spaces, and, in particular, as cells of a random tessellation of $\mathbb{R}^n$. We present results that pertain the geometry of such random polytopes in high dimensions, that is, as $n\to\infty$. In particular, we review their connection to the hyperplane or slicing conjecture, one of the major open problems in asymptotic geometric analysis.