Explicit IDP lattice polytopes in dimensions 7 and 12 have h*-vectors that are unimodal but not log-concave.
Unimodality questions for integrally closed lattice polytopes
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Examples of IDP lattice polytopes with non-log-concave $h^*$-vector
Explicit IDP lattice polytopes in dimensions 7 and 12 have h*-vectors that are unimodal but not log-concave.