Metric topologies over some categories of simple open regions in Euclidean space

TL;DR

This paper investigates how different metrics induce various topologies over spaces of simple open regions in Euclidean space, revealing equivalences and distinctions depending on the region class.

Contribution

It characterizes the relationships between multiple metrics and the topologies they induce over convex, union of convex, and star-shaped regions.

Findings

All five metrics induce the same topology over convex regions.

Metrics are ordered by fineness over convex unions, with homeomorphism-based being the finest.

Topologies are incomparable for star-shaped regions under certain metrics.

Abstract

What does it mean for a shape to change continuously? Over the space of convex regions, there is only one "reasonable" answer. However, over a broader class of regions, such as the class of star-shaped regions, there can be many different "reasonable" definitions of continuous shape change. We consider the relation between topologies induced by a number of metrics over a number of limited categories of open bounded regions in n-dimensional Euclidean space. Specifically, we consider a homeomorphism-based metric; the Hausdorff metric; the dual-Hausdorff metric; the symmetric difference metric; and the family of Wasserstein metrics; and the topologies that they induce over the space of convex regions; the space of convex regions and unions of two separated convex regions; and the space of star-shaped regions. We demonstrate that: Over the space of convex regions, all five metrics, and indeed any metric that satisfies two general well-behavedness constraints, induce the same topology. Over the space of convex regions and unions of two separated convex regions, these five metrics are all ordered by "strictly finer than" relations. In descending order of fineness, these are: the homeomorphism-based, the dual-Hausdorff, the Hausdorff, the Wasserstein, and the symmetric difference. Also, Wasserstein metrics are strictly ordered among themselves. Over the space of star-shaped regions, the topologies induced by the Hausdorff metric, the symmetric-difference metric, and the Wasserstein metrics are incomparable in terms of fineness.