Vol. 12, No. 7, 2019

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Closed geodesics on doubled polygons

Ian M. Adelstein and Adam Y. W. Fong

Vol. 12 (2019), No. 7, 1219–1227

We study 1k-geodesics, those closed geodesics that minimize on any subinterval of length Lk, 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(2n)-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 1k-geodesic.

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closed geodesics, regular polygons, billiard paths
Mathematical Subject Classification 2010
Primary: 53C20, 53C22
Received: 24 January 2019
Revised: 7 February 2019
Accepted: 18 February 2019
Published: 12 October 2019

Communicated by Frank Morgan
Ian M. Adelstein
Department of Mathematics
Yale University
New Haven, CT
United States
Adam Y. W. Fong
Department of Mathematics
Trinity College
Hartford, CT
United States