Multiset metrics on bounded spaces

TL;DR

This paper introduces five metrics on finite multisets in bounded metric spaces, generalizing the Hausdorff metric, and discusses their properties and potential applications.

Contribution

It presents new multiset metrics that extend the Hausdorff metric and analyzes their completeness and boundedness properties.

Findings

Four metrics are metrics iff the space is bounded.

All metrics are complete iff the space is complete.

The metrics generalize the Hausdorff metric.

Abstract

We discuss five simple functions on finite multisets of metric spaces. The first four are all metrics iff the underlying space is bounded and are complete metrics iff it is also complete. Two of them, and the fifth function, all generalise the usual Hausdorff metric on subsets. Some possible applications are also considered.