We consider the *n*-dimension Euclidean space **R**^{n}. If {*v*_{1}, ..., *v*_{n}} is a basis for **R**^{n}, then the set

*L* is in fact an abelian group, using the ordinary vector addition as operation.
One and the same lattice *L* may be generated by different bases, but the absolute value of the determinant of the vectors *v*_{i} is uniquely determined by *L*, and is denoted by d(*L*).
If one thinks of a lattice as dividing the whole of **R**^{n} into equal polyhedra, then d(*L*) is equal to the volume of this polyhedron.

The simplest example is the lattice **Z**^{n} of all points with integer coefficients; its determinant is 1.

Now let *S* be a convex subset of **R**^{n} that is symmetric with respect to the origin, meaning that *x* in *S* implies −*x* in *S*.
If *L* is a lattice in **R**^{n} and the volume of *S* is bigger than 2^{n}·d(*L*), then Minkowski's theorem states that *S* must contain at least 3 lattice points (the origin, another point, and its negative).