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In mathematics, Minkowski's theorem is the statement that every
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
in \mathbb^n which is symmetric with respect to the origin and which has
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
greater than 2^n contains a non-zero
integer point In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid ...
(meaning a point in \Z^n that is not the origin). The theorem was proved by Hermann Minkowski in 1889 and became the foundation of the branch of
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
called the
geometry of numbers Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in \mathbb R^n, and the study of these lattices provides fundamental informatio ...
. It can be extended from the integers to any
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
L and to any symmetric convex set with volume greater than 2^n\,d(L), where d(L) denotes the covolume of the lattice (the absolute value of the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of any of its bases).


Formulation

Suppose that is a
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
of
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
in the - dimensional
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
and is a
convex subset In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
of 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 In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid ...
of all points with
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
coefficients; its determinant is 1. For , the theorem claims that a convex figure in the Euclidean plane symmetric about the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...
and with
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...
greater than 4 encloses at least one lattice point in addition to the origin. The area bound is
sharp Sharp or SHARP may refer to: Acronyms * SHARP (helmet ratings) (Safety Helmet Assessment and Rating Programme), a British motorcycle helmet safety rating scheme * Self Help Addiction Recovery Program, a charitable organisation founded in 19 ...
: 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 . Proof of the \mathbb^2 case: Consider the map :f: S \to \mathbb^2/2L, \qquad (x,y) \mapsto (x \bmod 2, y \bmod 2) 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 In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
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 Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East ** Even language, a language spoken by the Evens * Odd and Even, a solitaire game w ...
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, :\tfrac\left(-p_1 + p_2\right) = \tfrac\left(-p_1 + p_1 + (2i, 2j)\right) = (i, j) is a point in . But 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 >\!\det(L) 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 2^n \det(L) is the covolume of the lattice 2L. * To obtain a proof for general lattices, it suffices to prove Minkowski's theorem only for \mathbb^n; this is because every full-rank lattice can be written as B\mathbb^n for some
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
B, and the properties of being convex and symmetric about the origin are preserved by linear transformations, while the covolume of B\mathbb^n is , \!\det(B), and volume of a body scales by exactly \frac under an application of B^.


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 L be a lattice. Then there is a x \in L \setminus \ with \, x\, _ \leq \left, \det(L)\^. In particular, by the standard comparison between l_2 and l_ norms, \, x\, _2 \leq \sqrt\, \left, \det(L)\^. Remarks: * The constant in the L^2 bound can be improved, for instance by taking the open ball of radius < l as C 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 (1,0), (0,n). * 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 i
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 \delta -LLL reduced basis b_1, \ldots, b_n for L has the property that \, b_1\, \leq \left(\frac\right)^ \det(L)^ ; see thes
lecture notes of Micciancio
for more on this. As explained in, proofs of bounds on the Hermite constant contain some of the key ideas in the LLL-reduction algorithm.


Applications to number theory


Primes that are sums of two squares

The difficult implication in
Fermat's theorem on sums of two squares In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be expressed as: :p = x^2 + y^2, with ''x'' and ''y'' integers, if and only if :p \equiv 1 \pmod. The prime numbers for which this is true ar ...
can be proven using Minkowski's bound on the shortest vector. Theorem: Every
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
with p \equiv 1 \mod 4 can be written as a sum of two
squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
. Additionally, the lattice perspective gives a computationally efficient approach to Fermat's theorem on sums of squares: First, recall that finding any nonzero vector with norm less than 2p in L, the lattice of the proof, gives a decomposition of p as a sum of two squares. Such vectors can be found efficiently, for instance using LLL-algorithm. In particular, if b_1, b_2 is a 3/4 -LLL reduced basis, then, by the property that \, b_1\, \leq (\frac)^ \text(B)^, \, b_1\, ^2 \leq \sqrt p < 2p. Thus, by running the LLL-lattice basis reduction algorithm with \delta = 3/4 , we obtain a decomposition of p as a sum of squares. Note that because every vector in L has norm squared a multiple of p, the vector returned by the LLL-algorithm in this case is in fact a shortest vector.


Lagrange's four-square theorem

Minkowski's theorem is also useful to prove
Lagrange's four-square theorem Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares. That is, the squares form an additive basis of order four. p = a_0^2 + a_1^2 + a_2^2 + a_ ...
, which states that every
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
can be written as the sum of the squares of four natural numbers.


Dirichlet's theorem on simultaneous rational approximation

Minkowski's theorem can be used to prove Dirichlet's theorem on simultaneous rational approximation.


Algebraic number theory

Another application of Minkowski's theorem is the result that every class in the
ideal class group In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...
of a
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
contains an integral ideal of
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
not exceeding a certain bound, depending on , called
Minkowski's bound In algebraic number theory, Minkowski's bound gives an upper bound of the norm of ideals to be checked in order to determine the class number of a number field In mathematics, an algebraic number field (or simply number field) is an extension ...
: the finiteness of the class number of an algebraic number field follows immediately.


Complexity theory

The complexity of finding the point guaranteed by Minkowski's theorem, or the closely related Blichfeldt's theorem, have been studied from the perspective of
TFNP In computational complexity theory, the complexity class TFNP is the class of total function problems which can be solved in nondeterministic polynomial time. That is, it is the class of function problems that are guaranteed to have an answer, and t ...
search problems. In particular, it is known that a computational analogue of Blichfeldt's theorem, a corollary of the proof of Minkowski's theorem, is PPP-complete. 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.


See also

*
Danzer set In geometry, a Danzer set is a set of points that touches every convex body of unit volume. Ludwig Danzer asked whether it is possible for such a set to have bounded density. Several variations of this problem remain unsolved. Density One way t ...
*
Pick's theorem In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1 ...
*
Dirichlet's unit theorem In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . The regulator is a pos ...
* Minkowski's second theorem * Ehrhart's volume conjecture


Further reading

* * * * * * * ( 996 with minor corrections * Wolfgang M. Schmidt.''Diophantine approximations and Diophantine equations'', Lecture Notes in Mathematics, Springer Verlag 2000. * *


External links

*Stevenhagen, Peter
''Number Rings''.
* *


References

{{DEFAULTSORT:Minkowski's Theorem Geometry of numbers Convex analysis Theorems in number theory Articles containing proofs Hermann Minkowski