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Oct 18, 2017 at 5:15 comment added user21574 Related to your question: Monotone toric symplectic manifolds have the unique toric structure (McDuff 2011) every smooth integral polytope is a facet of a monotone polytope, See the paper of Haase, Christian; Melnikov, Ilarion V. The reflexive dimension of a lattice polytope. Ann. Comb. 10 (2006), no. 2, 211–217.
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