Random points and lattice points in convex bodies

Random Points and Lattice Points in Convex Bodies

Assume K ⊂ Rd 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 ...

Random points and lattice points in convex bodies

We write K or Kd for the set of convex bodies in Rd, that is, compact convex sets with nonempty interior in Rd. Assume K ∈ K and x1, . . . , xn are random, independent points chosen according to the uniform distribution in K. The convex hull of these points, to be denoted by Kn, is called a random polytope inscribed in K. Thus Kn = [x1, . . . , xn] where [S] stands for the convex hull of the se...

Random Points, Convex Bodies, Lattices

Assume K is a convex body in R, 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 convX) 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 an...

Lattice points in large convex bodies , II

Covering convex bodies by cylinders and lattice points by flats ∗

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points o...