Random points and lattice points in convex bodies

Abstract

Assume $K \subset \mathbf {R}^d$ is a convex body and $X$ is a (large) finite subset of $K$? How many convex polytopes are there whose vertices belong to $X$? Is there a typical shape of such polytopes? How well does the maximal such polytope (which is actually the convex hull of $X$) approximate $K$? We are interested in these questions mainly in two cases. The first is when $X$ is a random sample of $n$ uniform, independent points from $K$. In this case motivation comes from Sylvester’s famous four-point problem and from the theory of random polytopes. The second case is when $X=K \cap \mathbf {Z}^d$ where $\mathbf {Z}^d$ is the lattice of integer points in $\mathbf {R}^d$ and the questions come from integer programming and geometry of numbers. Surprisingly (or not so surprisingly), the answers in the two cases are rather similar.