Integer points in convex polyhedra

The study of integer points in convex polyhedra

from the mathematical notation ${\displaystyle \mathbb {Z} }$ or Z for the set of integer numbers.

Properties

For a lattice Λ, Minkowski's theorem relates the number d(Λ) (the volume of a fundamental parallelepiped of the lattice) and the volume of a given symmetric convex set S to the number of lattice points contained in S.

The number of lattice points contained in a polytope all of whose vertices are elements of the lattice is described by the polytope's Ehrhart polynomial. Formulas for some of the coefficients of this polynomial involve d(Λ) as well.

Applications

Loop optimization

In certain approaches to loop optimization, the set of the executions of the loop body is viewed as the set of integer points in a polyhedron defined by loop constraints.

See also

• Convex lattice polytope
• Pick's theorem

References and notes