Curt McMullen showed that every compact orbit for the action of the diagonal group on the space of lattices contains a well-rounded lattice. We extend this to all closed orbits.
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Michael Levin, Uri Shapira, Barak Weiss, Closed orbits for the diagonal group and well-rounded lattices. Groups Geom. Dyn. 10 (2016), no. 4, pp. 1211–1225DOI 10.4171/GGD/380