Lattice points in rotated convex domains

Lattice points in large convex bodies , II

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...

Compact Weighted Composition Operators and Fixed Points in Convex Domains

We extend a classical result of Caughran/H. Schwartz and another recent result of Gunatillake by showing that if D is a bounded, convex domain in C, ψ : D → C is analytic and bounded away from zero toward the boundary of D, and φ : D → D is a holomorphic map such that the weighted composition operator Wψ,φ is compact on a holomorphic functional Hilbert space (containing the polynomial functions...

Equilibrium Points of Logarithmic Potentials on Convex Domains

Let D be a convex domain in C. Let ak > 0 be summable constants and let zk ∈ D. If the zk converge sufficiently rapidly to η ∈ ∂D from within an appropriate Stolz angle then the function ∑ ∞ k=1 ak/(z − zk) has infinitely many zeros in D. An example shows that the hypotheses on the zk are not redundant, and that two recently advanced conjectures are false. M.S.C. 2000 classification: 30D35, 31A...