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Encyclopedia > Minkowski's theorem

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Minkowski's theorem

Minkowski's theorem is a statement in the field of geometry of numbers about convex symmetric sets. It relates the number of contained lattice points to the volume of such a set.

We consider the n-dimension Euclidean space Rn. If {v1, ..., vn} is a basis for Rn, then the set

L = \left\{ \sum_{1 \le i \le n} a_i v_i \; | \; a_i \; \mbox{are integers} \right\} </math> is called a lattice in Rn.

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 vi is uniquely determined by L, and is denoted by d(L). If one thinks of a lattice as dividing the whole of Rn into equal polyhedra, then d(L) is equal to the volume of this polyhedron.

The simplest example is the lattice Zn of all points with integer coefficients; its determinant is 1.

Now let S be a convex subset of Rn which is symmetric with respect to the origin, meaning that x in S implies -x in S. If L is a lattice in Rn and the volume of S is bigger than 2n·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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