Lattice points in large convex planar domains of finite type

Lattice points in large convex bodies , II

Hölder estimates on convex domains of finite type

This article contains a natural and important application of the holomorphic support functions for convex domains of finite type in Cn constructed in [DiFo]. Namely, we use these functions to get ∂-solving Cauchy-Fantappié kernels for ∂-closed (0, q)-forms, such that the solutions given by them on bounded forms satisfy the best possible uniform Hölder estimates. More precisely we show: Theorem ...

Boundary Behaviour on Convex Domains of Finite Type

We describe the optimal approach regions for a theorem of Fatou type for H p functions on convex domains of nite type in C n. Moreover we show that the Nagel-Stein phenomenon also holds in this context.

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...