View source for Geometry of numbers

''geometric number theory''

The branch of number theory that studies number-theoretical problems by the use of geometric methods. Geometry of numbers in its proper sense was formulated by H. Minkowski in 1896 in his fundamental monograph [[#References|[1]]]. The starting point of this science, which subsequently became an independent branch of number theory, is the fact (already noted by Minkowski) that certain assertions which seem evident in the context of figures in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g0443501.png" />-dimensional Euclidean space have far-reaching consequences in number theory.

A fundamental and typical task of the geometry of numbers is the problem to determine the arithmetical minimum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g0443502.png" /> of some real function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g0443503.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g0443504.png" /> is the infimum of the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g0443505.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g0443506.png" /> runs through all the integral points (i.e. points with integer coordinates) that satisfy some supplementary condition (e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g0443507.png" />). In the most important special cases information on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g0443508.png" /> can be obtained from Minkowski's convex-body theorem, which may be formulated as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g0443509.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435010.png" />-dimensional convex body of volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435011.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435013.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435014.png" />; then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435015.png" /></td> </tr></table>

The quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435016.png" /> is useful in considering conditions of existence of solutions of the Diophantine inequality (cf. [[Diophantine approximations|Diophantine approximations]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435017.png" /></td> </tr></table>

This is a problem to which many problems in number theory can be reduced. The geometry of quadratic forms (cf. [[Quadratic form|Quadratic form]]) forms a separate chapter in the geometry of numbers.

Two general types of problems are distinguished in the geometry of numbers: the homogeneous and the inhomogeneous problem.

The homogeneous problem, which forms the subject of most studies in the geometry of numbers, deals with the homogeneous minima <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435018.png" /> of a distance function (cf. [[Ray function|Ray function]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435019.png" /> on a [[Lattice of points|lattice of points]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435020.png" />. The concept of a lattice (of points) is a fundamental one in the geometry of numbers. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435021.png" /> be linearly independent vectors in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435022.png" />-dimensional Euclidean space. The set of points

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435023.png" /></td> </tr></table>

when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435024.png" /> each run through all the integers in an independent manner, is known as the lattice (of points) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435025.png" /> with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435026.png" /> and determinant

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435027.png" /></td> </tr></table>

Let a distance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435028.png" /> and a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435029.png" /> with determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435030.png" /> be given in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435031.png" />. The greatest lower bound

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435032.png" /></td> </tr></table>

of the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435033.png" /> over the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435035.png" /> is called the minimum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435036.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435037.png" /> (or, more accurately, the homogeneous arithmetical minimum). The greatest lower bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435038.png" />, which may or may not be attained, is known to be attained by a bounded star body (cf. [[Star-like domain|Star-like domain]]), which is defined by the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435039.png" /></td> </tr></table>

In order to estimate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435040.png" /> from above one must calculate (or estimate) the constant of Hermite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435041.png" /> of the distance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435042.png" />, defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435043.png" /></td> </tr></table>

where the supremum is taken over the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435044.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435045.png" />-dimensional lattices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435046.png" />. There are relations between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435047.png" />, the critical determinant (see below) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435048.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435049.png" /> and (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435050.png" /> is a convex symmetric distance function) the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435051.png" /> of the densest lattice [[Packing|packing]] of the body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435052.png" />.

Let a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435053.png" /> and a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435054.png" /> with determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435055.png" /> be given in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435056.png" />. The lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435057.png" /> is called admissible for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435058.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435060.png" />-admissible, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435061.png" /> contains no non-zero points from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435062.png" />. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435063.png" /> with at least one admissible lattice is called a set of finite type; otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435064.png" /> is called a set of infinite type. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435065.png" /> be a set of finite type; the infimum

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435066.png" /></td> </tr></table>

of the set of determinants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435067.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435068.png" />-admissible lattices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435069.png" /> is called the critical determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435070.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435071.png" />. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435072.png" />-admissible lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435073.png" /> that satisfies the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435074.png" /></td> </tr></table>

is called a critical lattice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435075.png" />. For a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435076.png" /> of infinite type one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435077.png" />.

The calculation of the constant of Hermite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435078.png" /> of a distance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435079.png" /> is reduced to the computation of the critical determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435080.png" /> of the star body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435081.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435082.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435083.png" /></td> </tr></table>

The connection between the critical determinant and the density of the densest lattice packing is established by the following theorem of Blichfeldt. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435084.png" /> be an arbitrary set, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435085.png" /> be the corresponding set of differences (i.e. the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435086.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435087.png" />) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435088.png" /> be a lattice. For the arrangement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435089.png" />, i.e. for the family of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435090.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435091.png" />, to be a packing it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435092.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435093.png" />-admissible.

The density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435094.png" /> of the densest lattice packing of a bounded Lebesgue-measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435095.png" /> of measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435096.png" /> is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435097.png" /></td> </tr></table>

For an arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435098.png" /> and a Lebesgue-measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g04435099.png" /> of measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350100.png" /> that satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350101.png" /> the following inequality (another formulation of Blichfeldt's theorem) is valid:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350102.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350103.png" /> is a convex body that is symmetric with respect to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350104.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350105.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350106.png" /> is the density of the densest lattice packing of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350107.png" />. This means that in the case of a symmetric distance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350108.png" /> the computation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350109.png" /> is reduced to the computation of the densest lattice packing of the body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350110.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350111.png" />.

A very important statement in the geometry of numbers is Minkowski's convex-body theorem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350112.png" /> be a convex body that is symmetric with respect to the coordinate origin and of volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350113.png" />. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350114.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

In other words, a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350115.png" /> for which

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350116.png" /></td> </tr></table>

has a point distinct from zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350117.png" />.

Inequality (1) is known as the Minkowski inequality; it gives an estimate from below for the critical determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350118.png" /> of a convex body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350119.png" /> that is symmetric with respect to 0. In the general case this estimate cannot be improved. Equality is attained if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350120.png" />. Convex bodies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350121.png" /> that satisfy the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350122.png" /> are known as parallelohedra. They play an important role in the geometry of numbers and in mathematical crystallography (cf. [[Crystallography, mathematical|Crystallography, mathematical]]).

All applications of Minkowski's convex-body theorem are based on the fact that for a convex symmetric distance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350123.png" /> and an arbitrary lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350124.png" /> of determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350125.png" /> the following inequality is valid:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350126.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350127.png" /></td> </tr></table>

In particular, for the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350128.png" /> of integral points and the distance function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350129.png" /></td> </tr></table>

Minkowski's theorem on linear homogeneous forms is valid: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350130.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350131.png" /> be real numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350132.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350133.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350134.png" />. If

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350135.png" /></td> </tr></table>

then there exist integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350136.png" />, not all equal to zero, satisfying the system of linear inequalities

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350137.png" /></td> </tr></table>

Geometry of numbers also studies the successive minima of a distance function on a lattice. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350138.png" /> be a distance function, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350139.png" /> be a lattice and let there be given an index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350140.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350141.png" />; then the infimum of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350142.png" /> for which the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350143.png" /> contains at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350144.png" /> linearly independent points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350145.png" /> is said to be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350147.png" />-th successive minimum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350148.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350149.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350150.png" />. Here

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350151.png" /></td> </tr></table>

The estimate

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350152.png" /></td> </tr></table>

is valid. It is more difficult to estimate the magnitude

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350153.png" /></td> </tr></table>

from above; to do this, one must be able to compute, or to estimate from above, the quantity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350154.png" /></td> </tr></table>

where the supremum is over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350155.png" />-dimensional lattices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350156.png" />. The quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350157.png" /> is called the anomaly of the distance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350158.png" />, or the anomaly of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350159.png" />. The inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350160.png" /> is valid. The following theorem [[#References|[4]]] gives an estimate from above for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350161.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350162.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350163.png" />-dimensional distance function with anomaly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350164.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350165.png" /></td> </tr></table>

Examples have been constructed to show that this estimate cannot, generally speaking, be improved.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350166.png" /> is a convex symmetric distance function, it has been conjectured (the hypothesis on the anomaly of a convex body) that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350167.png" /></td> </tr></table>

Minkowski's second theorem on a convex body, making precise the first theorem, is valid. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350168.png" /> is a convex symmetric distance function and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350169.png" /> is a lattice, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350170.png" /></td> </tr></table>

where the convex body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350171.png" /> is defined by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350172.png" />. Minkowski's second theorem is valid [[#References|[4]]] independently of the hypothesis on the anomaly of a convex body.

The concept of successive minima and the fundamental results relevant to it (except for the last-named theorem) can be generalized from star bodies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350173.png" /> to arbitrary sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350174.png" /> [[#References|[9]]].

The following statement is an estimate from above of the critical determinant of a given set: For any Lebesgue-measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350175.png" /> of measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350176.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350177.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350178.png" /> is a star body that is symmetric with respect to zero, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350179.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350180.png" /></td> </tr></table>

All proofs of this theorem include some averaging of some function given on the space of lattices. The most natural proof is given by Siegel's mean-value theorem (see, e.g., [[#References|[12]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350181.png" /> be a Lebesgue-integrable function on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350182.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350183.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350184.png" /> be an invariant measure on the space of lattices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350185.png" /> with determinant 1. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350186.png" /> be the fundamental domain of this space, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350187.png" /></td> </tr></table>

As distinct from the estimate from below (1), estimates (2) and (3) are not the best possible (for more precise estimates see [[#References|[13]]]).

Estimates of the critical determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350188.png" /> of a given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350189.png" /> from below and from above yield estimates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350190.png" /> from above and from below, i.e. the solution (in a certain sense) of the homogeneous problem in the geometry of numbers. However, it is often important to know the exact value of the critical determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350191.png" /> for a given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350192.png" /> (e.g., for a norm body of a given algebraic number field). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350193.png" /> is a given bounded star body, then it is possible, in principle, to find an algorithm which permits one to reduce the problem of finding all critical lattices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350194.png" /> (and hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350195.png" /> as well) to a finite number of ordinary problems on the extrema of certain functions of several variables. However, this algorithm is realizable (in the present state of knowledge) only for convex bodies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350196.png" /> when the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350197.png" /> [[#References|[4]]].

Generally speaking, finding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350198.png" /> is much more difficult for unbounded star bodies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350199.png" />; this is clear by the isolation phenomenon of homogeneous arithmetical minima, which may be described as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350200.png" /> be a distance function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350201.png" />, and let the functional

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350202.png" /></td> </tr></table>

be given on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350203.png" /> of all lattices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350204.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350205.png" /> of possible values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350206.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350207.png" /> is called the Markov spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350208.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350209.png" /> has the isolation phenomenon if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350210.png" /> has isolated points. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350211.png" /> lies in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350212.png" />. If the star body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350213.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350214.png" />, is bounded, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350215.png" /></td> </tr></table>

For this reason the isolation phenomenon is possible for unbounded star bodies only (cf. [[#References|[4]]], Chapt. X). The most intensively studied case is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350216.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350217.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

A.N. Korkin and E.I. Zolotarev [[#References|[14]]] were the first to note the isolation phenomenon in this case (which was also the first case of the isolation phenomenon ever noted). A.A. Markov (see [[#References|[14]]]) proved in 1879 that the part of the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350218.png" /> to the right of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350219.png" /> is discrete, and has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350220.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350221.png" /> is an increasing sequence of positive integers with the following property: It is possible to find integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350222.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350223.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350224.png" /></td> </tr></table>

to each point of the spectrum (5) (the  "Markov spectrum"  in the narrow sense) there corresponds a unique (up to automorphisms [[#References|[4]]]) lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350225.png" />. The indefinite form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350226.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350227.png" />, is sometimes called the Markov form, while the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350228.png" /> is called a Markov chain. It is also known that to the left of some number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350229.png" /> the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350230.png" /> coincides with the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350231.png" />. The isolation phenomenon can be described in terms of admissible lattices (cf. [[#References|[9]]]), which generalizes this concept somewhat.

The inhomogeneous problem comprises the inhomogeneous Diophantine problems which play an important role in number theory; it forms an important branch of the geometry of numbers.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350232.png" /> be a distance function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350233.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350234.png" /> be a lattice of determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350235.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350236.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350237.png" /> be a point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350238.png" />. Consider the quantities

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350239.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350240.png" /></td> </tr></table>

where the infimum is over all points of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350241.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350242.png" />, while the supremum is over all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350243.png" />. The quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350244.png" /> is called the inhomogeneous arithmetical minimum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350245.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350246.png" />; this  "minimum"  need not be attained. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350247.png" /> is the greatest lower bound of the real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350248.png" /> having the following property: The arrangement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350249.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350250.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350251.png" /> satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350252.png" />, over the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350253.png" /> is a [[Covering|covering]], i.e.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350254.png" /></td> </tr></table>

For the distance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350255.png" /> one considers the following analogues of the Hermite constant:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350256.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350257.png" /></td> </tr></table>

where the infimum (supremum) is over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350258.png" />-dimensional lattices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350259.png" />. The quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350260.png" /> is usually trivial (cf. [[#References|[4]]]); if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350261.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350262.png" />, has a finite volume, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350263.png" /></td> </tr></table>

However, the inhomogeneous problem is connected with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350264.png" /> in one particular instance of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350265.png" /> which is of interest.

The hypothesis on the product of inhomogeneous linear forms may be stated as follows. Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350266.png" /></td> </tr></table>

then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350267.png" /></td> </tr></table>

Studies on this hypothesis and its analogues account for more than one half of all studies on the inhomogeneous problem in the geometry of numbers (cf. [[Minkowski hypothesis|Minkowski hypothesis]]).

In the general case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350268.png" /> is more informative than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350269.png" />. It is closely related to the value of the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350270.png" /> of the most economical covering by the body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350271.png" /> [[#References|[7]]], [[#References|[10]]]. In fact, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350272.png" /> is a distance function and if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350273.png" /> is bounded, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350274.png" /></td> </tr></table>

An important chapter of the inhomogeneous problems in the geometry of numbers is constituted by the so-called transference theorems for a given distance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350275.png" />, which are inequalities connecting the inhomogeneous minimum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350276.png" /> with the successive homogeneous minima <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350277.png" /> (or with the minima of the reciprocal function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350278.png" /> with respect to the reciprocal lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350279.png" />, etc., see [[#References|[4]]]). Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350280.png" /> be a convex symmetric distance function and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350281.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350282.png" />; then, for any lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350283.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350284.png" /></td> </tr></table>

There exist generalizations of the geometry of numbers to include spaces more general than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350285.png" /> and also to discrete sets more general than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350286.png" /> [[#References|[15]]], [[#References|[10]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Minkowski,   "Geometrie der Zahlen" , Chelsea, reprint  (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Minkowski,   "Diophantische Approximationen" , Chelsea, reprint  (1957)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Hancock,   "Development of the Minkowski geometry of numbers" , Macmillan  (1939)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.W.S. Cassels,   "An introduction to the geometry of numbers" , Springer  (1959)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.M. Gruber,   C.G. Lekkerkerker,   "Geometry of numbers" , North-Holland  (1987)  (Updated reprint)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L. Fejes Toth,   "Lagerungen in der Ebene, auf der Kugel und im Raum" , Springer  (1972)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  C.A. Rogers,   "Packing and covering" , Cambridge Univ. Press  (1964)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  O.-H. Keller,   "Geometrie der Zahlen" , ''Enzyklopaedie der math. Wissenschaften mit Einschluss ihrer Anwendungen'' , '''12'''  (1954)  (Heft 11, Teil III)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  E. Hlawka,   "Grundbegriffe der Geometrie der Zahlen"  ''Jahresber. Deutsch. Math.-Verein'' , '''57'''  (1954)  pp. 37–55</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  E.P. Baranovskii,   "Packings, coverings, partitionings and certain other distributions in spaces of constant curvature"  ''Progress Math.'' , '''9'''  (1971)  pp. 209–253  ''Itogi Nauk. Algebra Topol. Geom. 1967''  (1969)  pp. 189–225</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  J.F. Koksma,   "Diophantische Approximationen" , Springer  (1936)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  A.M. Macbeath,   C.A. Rogers,   "Siegel's mean value theorem in the geometry of numbers"  ''Proc. Cambridge Philos. Soc. (2)'' , '''54'''  (1958)  pp. 139–151</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  W. Schmidt,   "On the Minkowski–Hlawka theorem"  ''Illinois J. Math.'' , '''7'''  (1963)  pp. 18–23; 714</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  A.M. Markov,   "On binary quadratic forms with positive determinant"  ''Uspekhi Mat. Nauk'' , '''3''' :  5  (1948)  pp. 7–51  (In Russian)</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  K. Rogers,   H.P.F. Swinnerton-Dyer,   "The geometry of numbers over algebraic number fields"  ''Trans. Amer. Math. Soc.'' , '''88'''  (1958)  pp. 227–242</TD></TR></table>



====Comments====
A distance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350287.png" /> is a non-negative real-valued function on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350288.png" />-dimensional Euclidean space such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350289.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350290.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350291.png" /> it is called symmetric, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350292.png" /> it is called convex. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350293.png" /> is convex, it is required moreover that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350294.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350295.png" /> only.

The critical determinant of a lattice is also called the lattice constant. An arrangement is also called a set lattice. The inequality (3) is usually called the Minkowski–Hlawka theorem.

In recent years the geometry of numbers has become more geometric in character. The [[Covering and packing|covering and packing]] problems have been intensively studied, in particular the ball packing problem with its many relations to other areas such as coding, quantization of data, biology, metallurgy. Tilings have also attracted much interest; in particular Dirichlet–Voronoi (and Delone) tilings, which are of interest, for example, in geography, crystallography and computational geometry.

A tiling, or tesselation, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350296.png" /> is a family of sets (called tiles) such that their union covers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350297.png" /> and their interiors are mutually disjoint. A Dirichlet–Voronoi tiling is a tiling with as tiles sets of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350298.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350299.png" /> is a discrete point set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350300.png" />. Cf. [[#References|[5]]].

Other modern areas of the geometry of numbers are the theory of the zeta-function on lattices and (computational) reduction theory of quadratic forms and lattices.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Gruber,   "Geometry of numbers"  J. Tölke (ed.)  J.M. Wills (ed.) , ''Contributions to geometry'' , Birkhäuser  (1979)  pp. 186–225</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.H. Conway,   N.J.A. Sloane,   "Sphere packing, lattices and groups" , Springer  (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Erdös,   P.M. Gruber,   J. Hammer,   "Lattice points" , Longman  (1989)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S.S. Ryskov,   "The geometry of positive quadratic forms" , Amer. Math. Soc.  (1982)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.B. Malyshev,   Yu.G. Teterina,   "Investigations in number theory" , '''9''' , Leningrad  (1986)  (In Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  T.M. Thompson,   "From error-correcting codes through sphere packing to simple groups" , Math. Assoc. Amer.  (1983)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  G. Fejes Toth,   "New results in the theory of packing and covering"  P.M. Gruber (ed.)  J.M. Wills (ed.) , ''Convexity and its applications'' , Birkhäuser  (1983)  pp. 318–359</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  G.F. Voronoi,   "Collected works" , '''1–3''' , Kiev  (1952)  (In Russian)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  J.H.H. Chalk,   "Algebraic lattices"  P.M. Gruber (ed.)  J.M. Wills (ed.) , ''Convexity and its applications'' , Birkhäuser  (1983)  pp. 97–110</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  H. Minkowski,   "Gesammelte Abhandlungen" , '''1–2''' , Teubner  (1911)</TD></TR></table>