# Point distributions in compact metric spaces, II

## TL;DR

This paper investigates point distributions in compact metric spaces, providing bounds, invariance principles, and partitioning methods, thereby extending classical results and introducing probabilistic approaches for general spaces.

## Contribution

It generalizes Stolarsky's invariance principle to distance-invariant spaces and introduces a probabilistic invariance principle for arbitrary metric spaces.

## Key findings

- Bounds for sums of distances and discrepancies in rectifiable metric spaces
- Generalization of Stolarsky's invariance principle to distance-invariant spaces
- Probabilistic invariance principle for arbitrary metric spaces

## Abstract

We consider finite point subsets (distributions) in compact metric spaces. In the case of general rectifiable metric spaces, non-trivial bounds for sums of distances between points of distributions and for discrepancies of distributions in metric balls are given (Theorem 1.1). We generalize Stolarsky's invariance principle to distance-invariant spaces (Theorem 2.1). For arbitrary metric spaces, we prove a probabilistic invariance principle (Theorem 3.1). Furthermore, we construct equal-measure partitions of general rectifiable compact metric spaces into parts of small average diameter (Theorem 4.1). This version of the paper will be published in Mathematika