Minimizing closed geodesics on polygons and disks

TL;DR

This paper investigates 1/k geodesics on polygons and disks, developing new methods to analyze their minimizing properties, and reveals that certain doubled polygons have unbounded minimizing geodesics, with shortest geodesics approaching four times the diameter.

Contribution

The paper introduces novel techniques for studying minimizing properties of 1/k geodesics on doubled polygons and characterizes the behavior of shortest geodesics on doubled odd-gons.

Findings

Doubled polygons can have unbounded minimizing geodesics.

Shortest geodesic length on doubled odd-gons approaches 4 times the diameter.

New methods for analyzing 1/k geodesics on polygonal domains.

Abstract

In this paper we study 1/k geodesics, those closed geodesics that minimize on all subintervals of length $L/k$, where $L$ is the length of the geodesic. We develop new techniques to study the minimizing properties of these curves on doubled polygons, and demonstrate a sequence of doubled polygons whose closed geodesics exhibit unbounded minimizing properties. We also compute the length of the shortest closed geodesic on doubled odd-gons and show that this length approaches 4 times the diameter.