  Assume $K$ is a convex body in $R^d$, and $X$ is a (large) finite subset of
$K$. How many convex polytopes are there whose vertices come from $X$? What is
the typical shape of such a polytope? How well the largest such polytope (which
is actually $\conv X$) approximates $K$? 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 $K$ and is motivated by Sylvester's four-point problem,
and by the theory of random polytopes. The second case is when $X=K \cap Z^d$
where $Z^d$ is the lattice of integer points in $R^d$. Motivation comes from
integer programming and geometry of numbers. The two cases behave quite
similarly.
