Random Polytopes, Convex Bodies, and Approximation

Assume K ⊂ R is a convex body and Xn ⊂ K is a random sample of n uniform, independent points from K. The convex hull of Xn is a convex polytope Kn called random polytope inscribed in K. We are going to investigate various properties of this polytope: for instance how well it approximates K, or how many vertices and facets it has. It turns out that Kn is very close to the so called floating body inscribed in K with parameter 1/n. To show this we develop and use the technique of cap coverings and Macbeath regions. Its power will be illustrated, besides random polytopes, on several examples: floating bodies, lattice polytopes, and approximation problems.