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'''Geometry of numbers''' is the part of [[number theory]] which uses [[geometry]] for the study of [[algebraic number]]s. Typically, a [[ring of algebraic integers]] is viewed as a [[lattice (group)|lattice]] in <math>\mathbb R^n,</math> and the study of these lattices provides fundamental information on algebraic numbers.<ref>MSC classification, 2010, available at http://www.ams.org/msc/msc2010.html, Classification 11HXX.</ref>  {{harvs|txt|authorlink=Hermann Minkowski|first=Hermann|last= Minkowski|year=1896|ref1=https://mathweb.ucsd.edu/~b3tran/cgm/Minkowski_SpaceAndTime_1909.pdf}} initiated this line of research at the age of 26 in his work ''The Geometry of Numbers''.<ref>{{Cite book |last=Minkowski |first=Hermann |url=https://books.google.com/books?id=D-J9AgAAQBAJ&dq=Space+and+Time+Minkowski%E2%80%99s+Papers+on+Relativity&pg=PA1 |title=Space and Time: Minkowski's papers on relativity |date=2013-08-27 |publisher=Minkowski Institute Press |isbn=978-0-9879871-1-2 |language=en}}</ref>

{{Diophantine_approximation_graph.svg}}
The geometry of numbers has a close relationship with other fields of mathematics, especially [[functional analysis]] and [[Diophantine approximation]], the problem of finding [[rational number]]s <!-- or vectors with rational coordinates SIMPLIFY --> that <!-- accurately --> approximate an [[irrational number|irrational quantity]].<ref>Schmidt's books. {{Cite Geometric Algorithms and Combinatorial Optimization}}</ref>

==Minkowski's results==
{{Main article|Minkowski's theorem}}
Suppose that <math>\Gamma</math> is a [[Lattice (group)|lattice]] in <math>n</math>-dimensional Euclidean space <math>\mathbb{R}^n</math> and <math>K</math> is a convex centrally symmetric body.
[[Minkowski's theorem]], sometimes called Minkowski's first theorem, states that if <math>\operatorname{vol} (K)>2^n \operatorname{vol}(\mathbb{R}^n/\Gamma)</math>, then <math>K</math> contains a nonzero vector in <math>\Gamma</math>.

{{Main article|Minkowski's second theorem}}

The successive minimum <math>\lambda_k</math> is defined to be the [[Infimum|inf]] of the numbers <math>\lambda</math> such that <math>\lambda K</math> contains <math>k</math> linearly independent vectors of <math>\Gamma</math>.
Minkowski's theorem on [[successive minima]], sometimes called [[Minkowski's second theorem]], is a strengthening of his first theorem and states that<ref>Cassels (1971) p. 203</ref>
:<math>\lambda_1\lambda_2\cdots\lambda_n \operatorname{vol} (K)\le 2^n \operatorname{vol} (\mathbb{R}^n/\Gamma).</math>

==Later research in the geometry of numbers==
In 1930–1960 research on the geometry of numbers was conducted by many [[number theorist]]s (including [[Louis Mordell]], [[Harold Davenport]] and [[Carl Ludwig Siegel]]). In recent years, Lenstra, Brion, and Barvinok have developed combinatorial theories that enumerate the lattice points in some convex bodies.<ref>Grötschel et al., Lovász et al., Lovász, and Beck and Robins.</ref>

===Subspace theorem of W. M. Schmidt===
{{Main article|Subspace theorem}}
{{See also|Siegel's lemma|volume (mathematics)|determinant|parallelepiped}}
In the geometry of numbers, the [[subspace theorem]] was obtained by [[Wolfgang M. Schmidt]] in 1972.<ref>Schmidt, Wolfgang M. ''Norm form equations.'' Ann. Math. (2) '''96''' (1972), pp. 526–551.

See also Schmidt's books; compare Bombieri and Vaaler and also Bombieri and Gubler.</ref> It states that if ''n'' is a positive integer, and ''L''<sub>1</sub>,...,''L''<sub>''n''</sub> are [[linear independence|linearly independent]] [[linear]] [[algebraic form|forms]] in ''n'' variables with [[algebraic number|algebraic]] coefficients and if ε>0 is any given real number, then the non-zero integer points ''x'' in ''n'' coordinates with
:<math>|L_1(x)\cdots L_n(x)|<|x|^{-\varepsilon}</math>
lie in a finite number of [[linear subspace|proper subspaces]] of '''Q'''<sup>''n''</sup>.

==Influence on functional analysis==
{{Main article|normed vector space}}
{{See also|Banach space|F-space}}
Minkowski's geometry of numbers had a profound influence on [[functional analysis]]. Minkowski proved that symmetric convex bodies induce [[normed space|norms]] in finite-dimensional vector spaces. Minkowski's theorem was generalized to [[topological vector space]]s by [[Kolmogorov]], whose theorem states that the symmetric convex sets that are closed and bounded generate the topology of a [[Banach space]].<ref>For Kolmogorov's normability theorem, see Walter Rudin's ''Functional Analysis''. For more results, see Schneider, and Thompson and see Kalton et al.</ref>

Researchers continue to study generalizations to [[star-shaped set]]s and other [[convex set|non-convex set]]s.<ref>Kalton et al. Gardner</ref>

==References==
{{Reflist|2}}

==Bibliography==
* Matthias Beck, Sinai Robins. ''[[Computing the Continuous Discretely|Computing the continuous discretely: Integer-point enumeration in polyhedra]]'', [[Undergraduate Texts in Mathematics]], Springer, 2007.
* {{cite journal|author=Enrico Bombieri|author-link=Enrico Bombieri|author2=Vaaler, J.|title = On Siegel's lemma|journal = Inventiones Mathematicae|volume = 73|issue = 1|date = Feb 1983|pages = 11–32|doi = 10.1007/BF01393823|bibcode=1983InMat..73...11B|s2cid=121274024}}
* {{cite book
|author=Enrico Bombieri
|author-link=Enrico Bombieri
|author2=Walter Gubler
|name-list-style=amp
|title=Heights in Diophantine Geometry
|publisher=Cambridge U. P.
|year=2006}}
* [[J. W. S. Cassels]]. ''An Introduction to the Geometry of Numbers''. Springer Classics in Mathematics, Springer-Verlag 1997 (reprint of 1959 and 1971 Springer-Verlag editions).
* [[John Horton Conway]] and [[Neil Sloane|N. J. A. Sloane]], ''Sphere Packings, Lattices and Groups'', Springer-Verlag, NY, 3rd ed., 1998.
* R. J. Gardner, ''Geometric tomography,'' Cambridge University Press, New York, 1995. Second edition: 2006.
* [[Peter M. Gruber|P. M. Gruber]], ''Convex and discrete geometry,'' Springer-Verlag, New York, 2007.
* P. M. Gruber, J. M. Wills (editors), ''Handbook of convex geometry. Vol. A. B,'' North-Holland, Amsterdam, 1993.
* [[Martin Grötschel|M. Grötschel]], [[László Lovász|Lovász, L.]], [[Alexander Schrijver|A. Schrijver]]: ''Geometric Algorithms and Combinatorial Optimization'', Springer, 1988
* {{cite book
 | author = Hancock, Harris
 | title = Development of the Minkowski Geometry of Numbers
 | year = 1939
 | publisher = Macmillan}} (Republished in 1964 by Dover.)
* [[Edmund Hlawka]], Johannes Schoißengeier, Rudolf Taschner. ''Geometric and Analytic Number Theory''. Universitext. Springer-Verlag, 1991.
* {{citation
|last1=Kalton|first1=Nigel J.|author1-link=Nigel Kalton
|last2=Peck|first2=N. Tenney
|last3=Roberts|first3=James W.
| title = An F-space sampler
| series = London Mathematical Society Lecture Note Series, 89
| publisher = Cambridge University Press| location = Cambridge
| year = 1984| pages = xii+240| isbn = 0-521-27585-7 | mr = 0808777}}
* [[Gerrit Lekkerkerker|C. G. Lekkerkererker]]. ''Geometry of Numbers''. Wolters-Noordhoff, North Holland, Wiley. 1969.
* {{cite journal | author = Lenstra, A. K. | author-link = Arjen Lenstra | author2 = Lenstra, H. W. Jr. | author2-link = Hendrik Lenstra | author3 = Lovász, L. | author3-link = László Lovász | title = Factoring polynomials with rational coefficients | journal = [[Mathematische Annalen]] | volume = 261 | year = 1982 | issue = 4 | pages = 515–534 | hdl = 1887/3810 | doi = 10.1007/BF01457454 | mr = 0682664| s2cid = 5701340 | url = http://infoscience.epfl.ch/record/164484/files/nscan4.PDF }}
* [[László Lovász|Lovász, L.]]: ''An Algorithmic Theory of Numbers, Graphs, and Convexity'', CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986
* {{Springer|id=G/g044350|title=Geometry of numbers|first=A.V. |last=Malyshev}}
* {{Citation | last1=Minkowski | first1=Hermann | author1-link=Hermann Minkowski | title=Geometrie der Zahlen | url=https://archive.org/details/geometriederzahl00minkrich | publisher=R. G. Teubner | location=Leipzig and Berlin | mr=0249269 | year=1910 | jfm=41.0239.03 | access-date=2016-02-28}}
* [[Wolfgang M. Schmidt]]. ''Diophantine approximation''. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])
* {{cite book | last=Schmidt | first=Wolfgang M. | author-link=Wolfgang M. Schmidt | title=Diophantine approximations and Diophantine equations | series=Lecture Notes in Mathematics | volume=1467 | publisher=[[Springer-Verlag]] | year=1996 | edition=2nd | isbn=3-540-54058-X | zbl=0754.11020}}
* {{cite book | author = Siegel, Carl Ludwig | author-link = Carl Ludwig Siegel | title = Lectures on the Geometry of Numbers | url = https://archive.org/details/lecturesongeomet0000sieg | url-access = registration | year = 1989 | publisher = [[Springer-Verlag]]}}
* Rolf Schneider, ''Convex bodies: the Brunn-Minkowski theory,'' Cambridge University Press, Cambridge, 1993.
* Anthony C. Thompson, ''Minkowski geometry,'' Cambridge University Press, Cambridge, 1996.
* [[Hermann Weyl]]. Theory of reduction for arithmetical equivalence . Trans. Amer. Math. Soc. 48 (1940) 126–164. {{doi|10.1090/S0002-9947-1940-0002345-2}}
* Hermann Weyl. Theory of reduction for arithmetical equivalence. II . Trans. Amer. Math. Soc. 51 (1942) 203–231. {{doi|10.2307/1989946}}

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[[Category:Geometry of numbers| ]]