Minkowski's theorem
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In mathematics, Minkowski's theorem is the statement that any convex set in Rn which is symmetric with respect to the origin and with volume greater than 2n contains a non-zero lattice point. The theorem was proved by Hermann Minkowski in 1889 and became the foundation of the branch of number theory called the geometry of numbers.
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[edit] Formulation
Suppose that L is a lattice of determinant d(L) in the n-dimensional real vector space Rn and S is a convex subset of Rn that is symmetric with respect to the origin, meaning that if x is in S then −x is also in S. Minkowski's theorem states that if the volume of S is strictly greater than 2n d(L), then S must contain at least one lattice point other than the origin.[1]
[edit] Example
The simplest example of a lattice is the set Zn of all points with integer coefficients; its determinant is 1. For n = 2 the theorem claims that a convex figure in the plane symmetric about the origin and with area greater than 4 encloses at least one lattice point in addition to the origin. The area bound is sharp: if S is the interior of the square with vertices (±1, ±1) then S is symmetric and convex, has area 4, but the only lattice point it contains is the origin. This observation generalizes to every dimension n.
[edit] Proof
The following argument proves Minkowski's theorem for the special case of n = 2. It can be trivially generalized to arbitrary lattices in arbitrary dimensions.
Consider the map 
. Intuitively, this map cuts the plane into 2 by 2 squares, then stacks the squares on top of each other. Clearly f(S) has area ≤ 4. Suppose f were injective. Then each of the stacked squares would be non-overlapping, so f would be area-preserving, and the area of f(S) would be greater than 4, since S has area greater than 4. That is not the case, so f(p1) = f(p2) for some pair of points p1,p2 in S. Moreover, we know from the definition of f that p2 = p1 + (2i,2j) for some integers i and j, where i and j are not both zero.
Then since S is symmetric about the origin, − p1 is also a point in S. Since S is convex, the line segment between − p1 and p2 lies entirely in S, and in particular the midpoint of that segment lies in S. In other words,
lies in S. (i,j) is a lattice point, and is not the origin since i and j are not both zero, and so we have found the point we're looking for.
[edit] Applications
A corollary of this theorem is the fact that every class in the ideal class group of a number field K contains an integral ideal of norm not exceeding a certain bound, depending on K, called Minkowski's bound.
[edit] Notes
- ^ Since the set S is symmetric, it would then contain at least three lattice points: the origin 0 and a pair of points ±x, where x ∈ L \ 0.
[edit] See also
[edit] References
- Minkowski's theorem on PlanetMath
- Stevenhagen, Peter. Number Rings.
