Closed geodesics on doubled polygons

Abstract

We study $1∕k$-geodesics, those closed geodesics that minimize on any subinterval of length $L∕k$, where $L$ is the length of the geodesic. We investigate the existence and behavior of these curves on doubled polygons and show that every doubled regular $n$-gon admits a $1∕\left(2n\right)$-geodesic. For the doubled regular $p$-gons, with $p$ an odd prime, we conjecture that $k=2p$ is the minimum value for $k$ such that the space admits a $1∕k$-geodesic.