We study
-geodesics,
those closed geodesics that minimize on all subintervals of length
,
where
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
such that the space
admits a
-geodesic)
is unbounded. We also compute the length of the shortest closed
geodesic on doubled odd-gons and show that this length approaches
diameter.
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