  An interesting open problem in Ehrhart theory is to classify those lattice
polytopes having a unimodal $h^*$-vector. Although various sufficient
conditions have been found, necessary conditions remain a challenge. In this
paper, we consider integrally closed reflexive simplices and discuss an
operation that preserves reflexivity, integral closure, and unimodality of the
$h^*$-vector, providing one explanation for why unimodality occurs in this
setting. We also discuss the failure of proving unimodality in this setting
using weak Lefschetz elements.
