Motivated by work of the first author, this paper studies symplectic embedding problems of lagrangian products that are sufficiently symmetric. In general, lagrangian products arise naturally in the study of billiards. The main result of the paper is the rigidity of a large class of symplectic embedding problems of lagrangian products in any dimension. This is achieved by showing that the lagrangian products under consideration are symplectomorphic to toric domains, and by using the Gromov width and the cube capacity introduced by Gutt and Hutchings to obtain rigidity.
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