A lattice polytope is a convex polytope whose vertices belong to a specified point
lattice. After choosing a lattice basis, it
may be regarded as a convex polytope with integer
vertex coordinates. For a lattice polytope 
, the number of lattice points
in its nonnegative integer dilates 
is given by its Ehrhart
polynomial. A smooth polytope imposes an additional
basis condition on the directions of its incident edges
at each polytope vertex (Liu et al. 2026).
Lattice Polytope
See also
Ehrhart Polynomial, Point Lattice, Smooth PolytopeExplore with Wolfram|Alpha
References
Cox, D. A.; Little, J. B.; and Schenck, H. K. Toric Varieties. Providence, RI: Amer. Math. Soc., 2011. https://doi.org/10.1090/gsm/124.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.Cite this as:
Weisstein, Eric W. "Lattice Polytope." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LatticePolytope.html