A smooth polytope is a full-dimensional lattice polytope for which exactly 
edges meet at each polytope
vertex, and their primitive outgoing edge directions
form a lattice basis. Here 
is the dimension of the polytope,
and primitive means that a nonzero vector of the point
lattice is not a positive integer multiple, greater
than one, of another vector of the point
lattice. The condition is also called the Delzant condition (Liu et al. 2026).
Smoothness does not force every coefficient of the Ehrhart polynomial to be positive. Liu et al. (2026) determined the minimum possible
number of facets of a smooth 
-dimensional lattice polytope
with a negative Ehrhart polynomial coefficient.
There are no such examples in dimensions one and two. The minimum is 
in dimensions three and four, and 
for every 
. An explicit three-dimensional example with six facets
has
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