Minkowski's theorem

thumb|An [[ellipse|elliptical set in illustrating Minkowski's theorem. With sufficient size, it must contain other integer points than zero.]]

In mathematics, Minkowski's theorem is the statement that every convex set in <math>\mathbb{R}^n</math> which is symmetric with respect to the origin and which has volume greater than <math>2^n</math> contains a non-zero integer point (meaning a point in <math>\Z^n</math> that is not the origin). The theorem was proved by Hermann Minkowski in 1889 and became the foundation of the branch of number theory called the geometry of numbers. It can be extended from the integers to any lattice <math>L</math> and to any symmetric convex set with volume greater than <math>2^nd(L)</math>, where <math>d(L)</math> denotes the covolume of the lattice (the absolute value of the determinant of any of its bases).

Formulation

Suppose that is a lattice of determinant in the -dimensional real vector space <math>\mathbb{R}^n</math> and is a convex subset of <math>\mathbb{R}^n</math> that is symmetric with respect to the origin, meaning that if is in then is also in . Minkowski's theorem states that if the volume of is strictly greater than , then must contain at least one lattice point other than the origin. (Since the set is symmetric, it would then contain at least three lattice points: the origin 0 and a pair of points , where .)

Example

The simplest example of a lattice is the integer lattice <math>\mathbb{Z}^n</math> of all points with integer coefficients; its determinant is 1. For , the theorem claims that a convex figure in the Euclidean 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 is the interior of the square with vertices then is symmetric and convex, and has area 4, but the only lattice point it contains is the origin. This example, showing that the bound of the theorem is sharp, generalizes to hypercubes in every dimension .

Proof

The following argument proves Minkowski's theorem for the specific case of <math>L = \mathbb{Z}^2.</math>

Proof of the <math display = "inline"> \mathbb{Z}^2 </math> case: Consider the map

:<math>f: S \to \mathbb{R}^2/2L, \qquad (x,y) \mapsto (x \bmod 2, y \bmod 2)</math>

Intuitively, this map cuts the plane into 2 by 2 squares, then stacks the squares on top of each other. Clearly has area less than or equal to 4, because this set lies within a 2 by 2 square. Assume for a contradiction that could be injective, which means the pieces of cut out by the squares stack up in a non-overlapping way. Because is locally area-preserving, this non-overlapping property would make it area-preserving for all of , so the area of would be the same as that of , which is greater than 4. That is not the case, so the assumption must be false: is not injective, meaning that there exist at least two distinct points in that are mapped by to the same point: .

Because of the way was defined, the only way that can equal is for

to equal for some integers and , not both zero.

That is, the coordinates of the two points differ by two even integers.

Since is symmetric about the origin, is also a point in . Since is convex, the line segment between and lies entirely in , and in particular the midpoint of that segment lies in . In other words,

:<math>\tfrac{1}{2}\left(-p_1 + p_2\right) = \tfrac{1}{2}\left(-p_1 + p_1 + (2i, 2j)\right) = (i, j)</math>

is a point in . This point is an integer point, and is not the origin since and are not both zero.

Therefore, contains a nonzero integer point.

Remarks:

The argument above proves the theorem that any set of volume <math display = "inline"> >\!\det(L)</math> contains two distinct points that differ by a lattice vector. This is a special case of Blichfeldt's theorem.
The argument above highlights that the term <math display = "inline">2^n \det(L)</math> is the covolume of the lattice <math display = "inline">2L</math>.
To obtain a proof for general lattices, it suffices to prove Minkowski's theorem only for <math display = "inline">\mathbb{Z}^n</math>; this is because every full-rank lattice can be written as <math display = "inline">B\mathbb{Z}^n</math> for some linear transformation <math display = "inline">B</math>, and the properties of being convex and symmetric about the origin are preserved by linear transformations, while the covolume of <math display = "inline">B\mathbb{Z}^n</math> is <math display = "inline">|\!\det(B)|</math> and volume of a body scales by exactly <math display = "inline">\frac{1}{\det(B)}</math> under an application of <math display = "inline">B^{-1}</math>.

Applications

Bounding the shortest vector

Minkowski's theorem gives an upper bound for the length of the shortest nonzero vector. This result has applications in lattice cryptography and number theory.

Theorem (Minkowski's bound on the shortest vector): Let <math display="inline">L</math> be a lattice. Then there is a <math display="inline">x \in L \setminus \{0\}</math> with <math display="inline"> \|x\|_{\infty} \leq \left|\det(L)\right|^{1/n}</math>. In particular, by the standard comparison between <math display="inline">l_2</math> and <math display="inline">l_{\infty}</math> norms, <math display="inline"> \|x\|_2 \leq \sqrt{n}\, \left|\det(L)\right|^{1/n}</math>.

Remarks:

The constant in the <math display="inline">L^2</math> bound can be improved, for instance by taking the open ball of radius <math display="inline"> < l</math> as <math display="inline">C</math> in the above argument. The optimal constant is known as the Hermite constant.
The bound given by the theorem can be very loose, as can be seen by considering the lattice generated by <math display="inline">(1,0), (0,n)</math>. But it cannot be further improved in the sense that there exists a global constant <math>c</math> such that there exists an <math>n</math>-dimensional lattice <math>L</math> satisfying <math>\| x\|_2 \geq c {\sqrt{n\cdot \left|\det(L)\right|^{1/n}</math>for all <math>x \in L \setminus \{0\}</math>. Furthermore, such lattice can be self-dual.
Even though Minkowski's theorem guarantees a short lattice vector within a certain magnitude bound, finding this vector is in general a hard computational problem. Finding the vector within a factor guaranteed by Minkowski's bound is referred to as Minkowski's Vector Problem (MVP), and it is known that approximation SVP reduces to it using transference properties of the dual lattice. The computational problem is also sometimes referred to as HermiteSVP.
The LLL-basis reduction algorithm can be seen as a weak but efficiently algorithmic version of Minkowski's bound on the shortest vector. This is because a <math display="inline"> \delta </math>-LLL reduced basis <math display="inline"> b_1, \ldots, b_n </math> for <math display="inline"> L </math> has the property that <math display="inline"> \|b_1\| \leq \left(\frac{1}{ \delta - .25}\right)^{\frac{n-1}{4 \det(L)^{1/n} </math>; see these lecture notes of Micciancio for more on this. As explained in, It is also known that the computational analogue of Minkowski's theorem is in the class PPP, and it was conjectured to be PPP-complete.

References

External links

Stevenhagen, Peter. Number Rings.