Vol. 14, No. 1, 2021

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Minimizing closed geodesics on polygons and disks

Ian Adelstein, Arthur Azvolinsky, Joshua Hinman and Alexander Schlesinger

Vol. 14 (2021), No. 1, 11–52
Abstract

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 where the minimizing index (the smallest k such that the space admits a 1 k-geodesic) is unbounded. We also compute the length of the shortest closed geodesic on doubled odd-gons and show that this length approaches 4 diameter.

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Keywords
closed geodesics, regular polygons, billiards
Mathematical Subject Classification 2010
Primary: 53C22
Milestones
Received: 19 September 2019
Revised: 11 May 2020
Accepted: 5 July 2020
Published: 4 March 2021

Communicated by Frank Morgan
Authors
Ian Adelstein
Department of Mathematics
Yale University
New Haven, CT
United States
Arthur Azvolinsky
Department of Mathematics
Yale University
New Haven, CT
United States
Joshua Hinman
Department of Mathematics
Yale University
New Haven, CT
United States
Alexander Schlesinger
Department of Mathematics
Yale University
New Haven, CT
United States