  For a given lattice polytope, two fundamental problems within the field of
Ehrhart theory are to (1) determine if its (Ehrhart) $h^\ast$-polynomial is
unimodal and (2) to determine if its Ehrhart polynomial has only positive
coefficients. The former property of a lattice polytope is known as Ehrhart
unimodality and the latter property is known as Ehrhart positivity. These two
properties are often simultaneously conjectured to hold for interesting
families of lattice polytopes, yet they are typically studied in parallel. As
to answer a question posed at the 2017 Introductory Workshop to the MSRI
Semester on Geometric and Topological Combinatorics, the purpose of this note
is to show that there is no general implication between these two properties in
any dimension greater than two. To do so, we investigate these two properties
for families of well-studied lattice polytopes, assessing one property where
previously only the other had been considered. Consequently, new examples of
each phenomena are developed, some of which provide an answer to an open
problem in the literature. The well-studied families of lattice polytopes
considered include zonotopes, matroid polytopes, simplices of weighted
projective spaces, empty lattice simplices, smooth polytopes, and $s$-lecture
hall simplices.
