
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
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
contains a non-zero
integer point (meaning a point in
that is not the origin). The theorem was
proved by
Hermann Minkowski
Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in numb ...
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, "Math ...
called the
geometry of numbers. It can be extended from the integers to any
lattice and to any symmetric convex set with volume greater than
, where
denotes the
covolume
In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice poin ...
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 ...
of any of its bases).
Formulation
Suppose that is a
lattice 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 ...
in the -
dimensional real 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 ...
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 conve ...
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 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 language ...
coefficients; its determinant is 1. For , the theorem claims that a convex figure in the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
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 open su ...
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
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions ...
s in every dimension .
Proof
The following argument proves Minkowski's theorem for the specific case of .
Proof of the
case: Consider the map
:
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' ...
that could be
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...
, 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,
:
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
contains two distinct points that differ by a lattice vector. This is a special case of
Blichfeldt's theorem
Blichfeldt's theorem is a mathematical theorem in the geometry of numbers, stating that whenever a bounded set in the Euclidean plane has area A, it can be translated so that it includes at least \lceil A\rceil points of the integer lattice. Equiv ...
.
* The argument above highlights that the term
is the covolume of the lattice
.
* To obtain a proof for general lattices, it suffices to prove Minkowski's theorem only for
; this is because every full-rank lattice can be written as
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 pr ...
, and the properties of being convex and symmetric about the origin are preserved by linear transformations, while the covolume of
is
and volume of a body scales by exactly
under an application of
.
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
be a lattice. Then there is a
with
. In particular, by the standard comparison between
and
norms,
.
Remarks:
* The constant in the
bound can be improved, for instance by taking the open ball of radius
as
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
.
* 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 itusing
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
-LLL reduced basis
for
has the property that
; see thes
lecture notes of Miccianciofor 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 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 way ...
with
can be written as a sum of two
squares.
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
in
, the lattice of the proof, gives a decomposition of
as a sum of two squares. Such vectors can be found efficiently, for instance using
LLL-algorithm. In particular, if
is a
-LLL reduced basis, then, by the property that
,
. Thus, by running the LLL-lattice basis reduction algorithm with
, we obtain a decomposition of
as a sum of squares. Note that because every vector in
has norm squared a multiple of
, 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_3 ...
, 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 ...
of a
number field contains an
integral ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral doma ...
of
norm not exceeding a certain bound, depending on , called
Minkowski's bound: 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 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
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...
d to be PPP complete.
See also
*
Danzer set
*
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 ...
*
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 i ...
*
Minkowski's second theorem In mathematics, Minkowski's second theorem is a result in the geometry of numbers about the values taken by a norm on a lattice and the volume of its fundamental cell.
Setting
Let be a closed convex centrally symmetric body of positive finite volu ...
*
Ehrhart's volume conjecture
Further reading
*
*
*
*
*
*
* (
996 with minor corrections
Year 996 ( CMXCVI) was a leap year starting on Wednesday (link will display the full calendar) of the Julian calendar.
Events
By place
Japan
* February - Chotoku Incident: Fujiwara no Korechika and Takaie shoot an arrow at Retired Emp ...
*
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