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A long-standing open conjecture in combinatorics asserts that a Gorensteinlattice polytope with the integer decomposition property (IDP) has a unimodal(Ehrhart) $h^\ast$-polynomial. This conjecture can be viewed as a strengtheningof a previously disproved conjecture which stated that any Gorenstein latticepolytope has a unimodal $h^\ast$-polynomial. The first counterexamples tounimodality for Gorenstein lattice polytopes were given in even dimensionsgreater than five by Musta{\c{t}}{\v{a}} and Payne, and this was extended toall dimensions greater than five by Payne. While there exist numerous examplesin support of the conjecture that IDP reflexives are $h^\ast$-unimodal, itsvalidity has not yet been considered for families of reflexive latticesimplices that closely generalize Payne's counterexamples. The main purpose ofthis work is to prove that the former conjecture does indeed hold for a naturalgeneralization of Payne's examples. The second purpose of this work is toextend this investigation to a broader class of lattice simplices, for which wepresent new results and open problems.

Detecting the Integer Decomposition Property and Ehrhart Unimodality in Reflexive Simplices