Point distributions in compact metric spaces

TL;DR

This paper investigates point distributions in compact metric spaces, establishing bounds on distances and discrepancies, generalizing Stolarsky's invariance principle, and constructing optimal partitions with minimal average diameter.

Contribution

It generalizes Stolarsky's invariance principle to a broader class of metric spaces and introduces probabilistic invariance principles and optimal partition constructions.

Findings

Derived bounds for sums of distances in metric spaces

Generalized Stolarsky's invariance principle to distance-invariant spaces

Constructed partitions with minimal average diameter

Abstract

We consider finite point subsets (distributions) in compact metric spaces. Non-trivial bounds for sums of distances between points of distributions and for discrepancies of distributions in metric balls are given in the case of general rectifiable metric spaces. We generalize Stolarsky's invariance principle to distance-invariant spaces, and for arbitrary metric spaces we prove a probabilistic invariance principle. Furthermore, we construct partitions of general rectifiable compact metric spaces into subsets of equal measure with minimum average diameter.