2.
is half the sum of the contents of the -dimensional faces of .
3. .
Let
denote the sum of the lattice lengths of the edges of , then the case corresponds to Pick's theorem,
(3)
Let
denote the sum of the lattice volumes of the two-dimensional faces of , then the case gives
(4)
where a rather complicated expression is given by Pommersheim (1993), since can unfortunately not be interpreted
in terms of the edges of . The Ehrhart polynomial of the tetrahedron with vertices
at (0, 0, 0), (,
0, 0), (0, ,
0), (0, 0, )
is
Even smooth polytopes can have negative Ehrhart coefficients. Liu et al. (2026) determined the least number of facets
for such a -dimensional
lattice polytope. There are no examples for or . The minimum is for or , and for every . In dimension three they gave a six-facet example with
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Springer, 2007.Ehrhart, E. "Sur une problème de géométrie
diophantine linéaire." J. reine angew. Math.227, 1-29,
1967.Ellis-Monaghan, J. A. and Merino, C. "Graph Polynomials
and Their Applications II: Interrelations and Interpretations." 28 Jun 2008.
https://arxiv.org/abs/0806.4699.Gardner,
M. The
Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University
of Chicago Press, p. 215, 1984.Liu, F.; Tang, J.; Tao, S.; and
Zhang, Z. "Ehrhart Positivity of Smooth Polytopes with Few Facets." 20
Oct 2026. https://arxiv.org/abs/2610.27819.Macdonald,
I. G. "The Volume of a Lattice Polyhedron." Proc. Camb. Phil. Soc.59,
719-726, 1963.McMullen, P. "Valuations and Euler-Type Relations
on Certain Classes of Convex Polytopes." Proc. London Math. Soc.35,
113-135, 1977.Pommersheim, J. "Toric Varieties, Lattices Points,
and Dedekind Sums." Math. Ann.295, 1-24, 1993.Reeve,
J. E. "On the Volume of Lattice Polyhedra." Proc. London Math.
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on the Volume of Lattice Polyhedra." Proc. London Math. Soc.34,
57-62, 1959.
of an 
-dimensional lattice polytope 
counts the lattice
points in its integer dilates. If the point
lattice is 
,
then
.
Ehrhart's theorem states that this counting function is a polynomial
in 
of degree 
with rational coefficients,
is the content of 
.
is half the sum of the contents of the 
-dimensional faces of 
.
.
denote the sum of the lattice lengths of the edges of 
, then the case 
corresponds to Pick's theorem,
denote the sum of the lattice volumes of the two-dimensional faces of 
, then the case 
gives
can unfortunately not be interpreted
in terms of the edges of 
. The Ehrhart polynomial of the tetrahedron with vertices
at (0, 0, 0), (
,
0, 0), (0, 
,
0), (0, 0, 
)
is
![l_Delta(k)=1/6abck^3+1/4(ab+ac+bc+d)k^2+[1/(12)((ac)/b+(bc)/a+(ab)/c+(d^2)/(abc))+1/4(a+b+c+A+B+C)-As((bc)/d,(aA)/d)-Bs((ac)/d,(bB)/d)-Cs((ab)/d,(cC)/d)]k+1,](/images/equations/EhrhartPolynomial/NumberedEquation5.svg)
is a Dedekind sum, 
, 
, 
(here, GCD is the greatest
common divisor), and 
(Pommersheim 1993).
-dimensional
lattice polytope. There are no examples for 
or 
. The minimum is 
for 
or 
, and 
for every 
. In dimension three they gave a six-facet example with
