# Shortest paths in arbitrary plane domains

## TL;DR

This paper proves the existence and uniqueness of shortest paths in arbitrary plane domains, showing they can be homotopically deformed from any given path while remaining inside the domain, even without finite Euclidean length.

## Contribution

It generalizes shortest path existence and uniqueness results to arbitrary plane domains, introducing new notions of shortest paths and homotopies.

## Key findings

- Unique shortest paths exist in arbitrary plane domains.
- Shortest paths can be homotopically deformed from any given path within the domain.
- Results extend previous work from simply connected domains to general domains.

## Abstract

Let $\Omega$ be a connected open set in the plane and $\gamma: [0,1] \to \overline{\Omega}$ a path such that $\gamma((0,1)) \subset \Omega$. We show that the path $\gamma$ can be ``pulled tight'' to a unique shortest path which is homotopic to $\gamma$, via a homotopy $h$ with endpoints fixed whose intermediate paths $h_t$, for $t \in [0,1)$, satisfy $h_t((0,1)) \subset \Omega$. We prove this result even in the case when there is no path of finite Euclidean length homotopic to $\gamma$ under such a homotopy. For this purpose, we offer three other natural, equivalent notions of a ``shortest'' path. This work generalizes previous results for simply connected domains with simple closed curve boundaries.