  Within the intersection of Ehrhart theory, commutative algebra, and algebraic
geometry lie lattice polytopes. Ehrhart theory is concerned with lattice point
enumeration in dilates of polytopes; lattice polytopes provide a sandbox in
which to test many conjectures in commutative algebra; and many properties of
projectively normal toric varieties in algebraic geometry are encoded through
corresponding lattice polytopes. In this article we focus on reflexive
simplices and work to identify when these have the integer decomposition
property (IDP), or equivalently, when certain weighted projective spaces are
projectively normal. We characterize the reflexive, IDP simplices whose
associated weighted projective spaces have one projective coordinate with
weight fixed to unity and for which the remaining coordinates can assume one of
three distinct weights. We show that several subfamilies of such reflexive
simplices have unimodal $h^\ast$-polynomials, thereby making progress towards
conjectures and questions of Stanley, Hibi-Ohsugi, and others regarding the
unimodality of their $h^\ast$-polynomials. We also provide computational
results and introduce the notion of reflexive stabilizations to explore the
(non-)ubiquity of reflexive simplices that are simultaneously IDP and
$h^\ast$-unimodal.
