Optimal homeomorphisms between closed curves

TL;DR

This paper proves the existence of optimal homeomorphisms between closed curves that precisely realize the natural pseudo-distance, a measure of dissimilarity between topological spaces with functions.

Contribution

It provides the first proof of the existence of such optimal homeomorphisms for closed curves, advancing the understanding of natural pseudo-distance in topological data analysis.

Findings

Existence of optimal homeomorphisms between closed curves established

Optimal homeomorphisms induce changes equal to the natural pseudo-distance

Advances the theoretical foundation of measuring topological dissimilarity

Abstract

The concept of natural pseudo-distance has proven to be a powerful tool for measuring the dissimilarity between topological spaces endowed with continuous real-valued functions. Roughly speaking, the natural pseudo-distance is defined as the infimum of the change of the functions' values, when moving from one space to the other through homeomorphisms, if possible. In this paper, we prove the first available result about the existence of optimal homeomorphisms between closed curves, i.e. inducing a change of the function that equals the natural pseudo-distance.