Abstract: We show that the following classes of lattice polytopes have unimodular covers, in dimension three: parallelepipeds, smooth centrally symmetric polytopes, and Cayley sums Cay (P, Q) where the normal fan of refines that of Q refines that of P. This improves results of Beck et al. (2018) and Haase et al. (2008) where the last two classes were shown to be IDP.

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