Quasi-uniform designs with optimal and near-optimal uniformity constant

TL;DR

This paper introduces a new approach to constructing quasi-uniform designs in compact subsets of Euclidean space, providing optimal bounds on their uniformity constants and extending the construction method for greater flexibility.

Contribution

It derives a lower bound on the uniformity constant for nested designs and presents a greedy construction that achieves this bound, with extensions for increased design flexibility.

Findings

A lower bound on the uniformity constant is established.

A greedy algorithm achieves the optimal uniformity bound.

Extensions allow more flexible design constructions.

Abstract

A design is a collection of distinct points in a given set $X$, which is assumed to be a compact subset of $R^d$, and the mesh-ratio of a design is the ratio of its fill distance to its separation radius. The uniformity constant of a sequence of nested designs is the smallest upper bound for the mesh-ratios of the designs. We derive a lower bound on this uniformity constant and show that a simple greedy construction achieves this lower bound. We then extend this scheme to allow more flexibility in the design construction.