In mathematics, Minkowski's second theorem is a result in 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 ...
about the values taken by a
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 envir ...
on a lattice and the volume of its fundamental cell.
Setting
Let be a
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
centrally symmetric
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
body of positive finite volume in -dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
. The ''gauge'' or ''distance''
Minkowski functional
In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space.
If K is a subset of a real or complex vector space X, the ...
attached to is defined by
Conversely, given a norm on we define to be
Let be 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 ornam ...
in . The successive minima of or on are defined by setting the -th successive minimum to be the
infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
of the numbers such that contains linearly-independent vectors of . We have .
Statement
The successive minima satisfy
[Siegel (1989) p.57]
Proof
A basis of linearly independent lattice vectors can be defined by .
The lower bound is proved by considering the convex
polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
with vertices at , which has an interior enclosed by and a volume which is times an integer multiple of a
primitive cell
In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessaril ...
of the lattice (as seen by scaling the polytope by along each basis vector to obtain
-simplices with lattice point vectors).
To prove the upper bound, consider functions sending points in
to the centroid of the subset of points in
that can be written as
for some real numbers
. Then the coordinate transform
has a Jacobian determinant
. If
and
are in the
interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
of
and
(with
) then
with
, where the inclusion in
(specifically the interior of
) is due to convexity and symmetry. But lattice points in the interior of
are, by definition of
, always expressible as a linear combination of
, so any two distinct points of
cannot be separated by a lattice vector. Therefore,
must be enclosed in a primitive cell of the lattice (which has volume
), and consequently
.
References
*
*
*
*
* {{cite book , first=Carl Ludwig , last=Siegel , author-link=Carl Ludwig Siegel , title=Lectures on the Geometry of Numbers , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, year=1989 , isbn=3-540-50629-2 , editor=Komaravolu S. Chandrasekharan , editor-link=Komaravolu S. Chandrasekharan , zbl=0691.10021
Hermann Minkowski