Asymptotics of best-packing on rectifiable sets

Asymptotics of Best-packing on Rectifiable Sets

We investigate the asymptotic behavior, as N grows, of the largest minimal pairwise distance of N points restricted to an arbitrary compact rectifiable set embedded in Euclidean space, and we find the limit distribution of such optimal configurations. For this purpose, we compare best-packing configurations with minimal Riesz s-energy configurations and determine the s-th root asymptotic behavi...

Asymptotics of weighted best-packing on rectifiable sets

We investigate the asymptotic behaviour, as N grows, of the largest minimal weighted pairwise distance between N points restricted to a rectifiable compact set embedded in Euclidean space, and we find the limit distribution of asymptotically optimal configurations. Bibliography: 23 titles. The classical best-packing problem is the problem of finding a configuration of N points on a given compac...

Information Theory on Rectifiable Sets

Entropy and—maybe even more so—mutual information are invaluable tools for analyzing properties of probability distributions, especially in coding theory. While there are general definitions for both concepts available for arbitrary probability distributions, these tend to be hard to work with and the literature (e.g., [CT06]) focuses on either discrete, or continuous random variables. In this ...

Asymptotics for Discrete Weighted Minimal Riesz Energy Problems on Rectifiable Sets

Given a compact d-rectifiable set A embedded in Euclidean space and a distribution ρ(x) with respect to d-dimensional Hausdorff measure on A, we address the following question: how can one generate optimal configurations of N points on A that are “well-separated” and have asymptotic distribution ρ(x) as N → ∞? For this purpose we investigate minimal weighted Riesz energy points, that is, points...

Non-rectifiable Limit Sets of Dimension One