We consider the ndimension 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} which 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).
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