Geometry of numbers

<h2 id="minkowskis-results">Minkowski's results</h2>
Main article: <a href="/facts/Minkowski%2527s_theorem/G2A8J5k6">Minkowski's theorem</a>
Suppose that 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
 is a <a href="/facts/Lattice_(group)/2ls9Iv9N">lattice</a> in 
 
 
 
 n
 
 
 {\displaystyle n}
 
-dimensional Euclidean space 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 and 
 
 
 
 K
 
 
 {\displaystyle K}
 
 is a convex centrally symmetric body.
<a href="/facts/Minkowski%2527s_theorem/G2A8J5k6">Minkowski's theorem</a>, sometimes called Minkowski's first theorem, states that if 
 
 
 
 vol
 ⁡
 (
 K
 )
 >
 
 2
 
 n
 
 
 vol
 ⁡
 (
 
 
 R
 
 
 n
 
 
 
 /
 
 Γ
 )
 
 
 {\displaystyle \operatorname {vol} (K)>2^{n}\operatorname {vol} (\mathbb {R} ^{n}/\Gamma )}
 
, then 
 
 
 
 K
 
 
 {\displaystyle K}
 
 contains a nonzero vector in 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
.

Main article: <a href="/facts/Minkowski%2527s_second_theorem/WsnWzBMx">Minkowski's second theorem</a>
The successive minimum 
 
 
 
 
 λ
 
 k
 
 
 
 
 {\displaystyle \lambda _{k}}
 
 is defined to be the <a href="/facts/Infimum/oXCKVopz">inf</a> of the numbers 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 such that 
 
 
 
 λ
 K
 
 
 {\displaystyle \lambda K}
 
 contains 
 
 
 
 k
 
 
 {\displaystyle k}
 
 linearly independent vectors of 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
.
Minkowski's theorem on <a href="/facts/Successive_minima/WsnWzBMx">successive minima</a>, sometimes called <a href="/facts/Minkowski%2527s_second_theorem/WsnWzBMx">Minkowski's second theorem</a>, is a strengthening of his first theorem and states that<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

λ
          
            1
          
        
        
          λ
          
            2
          
        
        ⋯
        
          λ
          
            n
          
        
        vol
        ⁡
        (
        K
        )
        ≤
        
          2
          
            n
          
        
        vol
        ⁡
        (
        
          
            R
          
          
            n
          
        
        
          /
        
        Γ
        )
        .
      
    
    {\displaystyle \lambda _{1}\lambda _{2}\cdots \lambda _{n}\operatorname {vol} (K)\leq 2^{n}\operatorname {vol} (\mathbb {R} ^{n}/\Gamma ).}

<h2 id="later-research-in-the-geometry-of-numbers">Later research in the geometry of numbers</h2>
In 1930–1960 research on the geometry of numbers was conducted by many <a href="/facts/Number_theorist/zhoa7zrc">number theorists</a> (including <a href="/facts/Louis_Mordell/L2Qpjnnz">Louis Mordell</a>, <a href="/facts/Harold_Davenport/aceBKgHd">Harold Davenport</a> and <a href="/facts/Carl_Ludwig_Siegel/I91aMunI">Carl Ludwig Siegel</a>). In recent years, Lenstra, Brion, and Barvinok have developed combinatorial theories that enumerate the lattice points in some convex bodies.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<h3>Subspace theorem of W. M. Schmidt</h3>
Main article: <a href="/facts/Subspace_theorem/Y8vcqDzH">Subspace theorem</a>
See also: <a href="/facts/Siegel%2527s_lemma/tTKMqWY1">Siegel's lemma</a>, <a href="/facts/Volume_(mathematics)/aeAXCBUd">volume (mathematics)</a>, <a href="/facts/Determinant/eGYqJ2TF">determinant</a>, and <a href="/facts/Parallelepiped/Qn7cDgSr">parallelepiped</a>
In the geometry of numbers, the <a href="/facts/Subspace_theorem/Y8vcqDzH">subspace theorem</a> was obtained by <a href="/facts/Wolfgang_M._Schmidt/5uFcNFt5">Wolfgang M. Schmidt</a> in 1972.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> It states that if n is a positive integer, and L1,...,Ln are <a href="/facts/Linear_independence/8JrHfIIa">linearly independent</a> <a href="/facts/Linear/fChtg3KC">linear</a> <a href="/facts/Algebraic_form/WUHL7Kct">forms</a> in n variables with <a href="/facts/Algebraic_number/MkH1PzTK">algebraic</a> coefficients and if ε>0 is any given real number, then the non-zero integer points x in n coordinates with

|
 
 
 L
 
 1
 
 
 (
 x
 )
 ⋯
 
 L
 
 n
 
 
 (
 x
 )
 
 |
 
 <
 
 |
 
 x
 
 
 |
 
 
 −
 ε
 
 
 
 
 {\displaystyle |L_{1}(x)\cdots L_{n}(x)|<|x|^{-\varepsilon }}

lie in a finite number of <a href="/facts/Linear_subspace/2wtKTgHL">proper subspaces</a> of Qn.

<h2 id="influence-on-functional-analysis">Influence on functional analysis</h2>
Main article: <a href="/facts/Normed_vector_space/rmDljfyE">normed vector space</a>
See also: <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> and <a href="/facts/F-space/NBCcYfcU">F-space</a>
Minkowski's geometry of numbers had a profound influence on <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>. Minkowski proved that symmetric convex bodies induce <a href="/facts/Normed_space/rmDljfyE">norms</a> in finite-dimensional vector spaces. Minkowski's theorem was generalized to <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a> by <a href="/facts/Kolmogorov/bEKwtrs6">Kolmogorov</a>, whose theorem states that the symmetric convex sets that are closed and bounded generate the topology of a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
Researchers continue to study generalizations to <a href="/facts/Star-shaped_set/gMTBXf96">star-shaped sets</a> and other <a href="/facts/Convex_set/vdAuJRJl">non-convex sets</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<h2 id="bibliography">Bibliography</h2>
<ul><li>Matthias Beck, Sinai Robins. <a href="/facts/Computing_the_Continuous_Discretely/qHCw8VCT">Computing the continuous discretely: Integer-point enumeration in polyhedra</a>, <a href="/facts/Undergraduate_Texts_in_Mathematics/MTjjHqWD">Undergraduate Texts in Mathematics</a>, Springer, 2007.</li>
<li><a href="/facts/Enrico_Bombieri/MU6xU5vr">Enrico Bombieri</a>; Vaaler, J. (Feb 1983). "On Siegel's lemma". Inventiones Mathematicae. 73 (1): 11–32. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1983InMat..73...11B">1983InMat..73...11B</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01393823">10.1007/BF01393823</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:121274024">121274024</a>.</li>
<li><a href="/facts/Enrico_Bombieri/MU6xU5vr">Enrico Bombieri</a> & Walter Gubler (2006). Heights in Diophantine Geometry. Cambridge U. P.</li>
<li><a href="/facts/J._W._S._Cassels/AC33eVoA">J. W. S. Cassels</a>. An Introduction to the Geometry of Numbers. Springer Classics in Mathematics, Springer-Verlag 1997 (reprint of 1959 and 1971 Springer-Verlag editions).</li>
<li><a href="/facts/John_Horton_Conway/g3PA1zoK">John Horton Conway</a> and <a href="/facts/Neil_Sloane/kLWw29Me">N. J. A. Sloane</a>, Sphere Packings, Lattices and Groups, Springer-Verlag, NY, 3rd ed., 1998.</li>
<li>R. J. Gardner, Geometric tomography, Cambridge University Press, New York, 1995. Second edition: 2006.</li>
<li><a href="/facts/Peter_M._Gruber/8hek0VEV">P. M. Gruber</a>, Convex and discrete geometry, Springer-Verlag, New York, 2007.</li>
<li>P. M. Gruber, J. M. Wills (editors), Handbook of convex geometry. Vol. A. B, North-Holland, Amsterdam, 1993.</li>
<li><a href="/facts/Martin_Gr%25C3%25B6tschel/9I4LlIhv">M. Grötschel</a>, <a href="/facts/L%25C3%25A1szl%25C3%25B3_Lov%25C3%25A1sz/X0VB38Ee">Lovász, L.</a>, <a href="/facts/Alexander_Schrijver/eJHJNRGz">A. Schrijver</a>: Geometric Algorithms and Combinatorial Optimization, Springer, 1988</li>
<li>Hancock, Harris (1939). Development of the Minkowski Geometry of Numbers. Macmillan. (Republished in 1964 by Dover.)</li>
<li><a href="/facts/Edmund_Hlawka/kRtwlzbA">Edmund Hlawka</a>, Johannes Schoißengeier, Rudolf Taschner. Geometric and Analytic Number Theory. Universitext. Springer-Verlag, 1991.</li>
<li><a href="/facts/Nigel_Kalton/QG63W3jY">Kalton, Nigel J.</a>; Peck, N. Tenney; Roberts, James W. (1984), An F-space sampler, London Mathematical Society Lecture Note Series, 89, Cambridge: Cambridge University Press, pp. xii+240, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-27585-7, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0808777">0808777</a></li>
<li><a href="/facts/Gerrit_Lekkerkerker/jXYFfu65">C. G. Lekkerkererker</a>. Geometry of Numbers. Wolters-Noordhoff, North Holland, Wiley. 1969.</li>
<li><a href="/facts/Arjen_Lenstra/GG04Nhd9">Lenstra, A. K.</a>; <a href="/facts/Hendrik_Lenstra/8XQ4Izlf">Lenstra, H. W. Jr.</a>; <a href="/facts/L%25C3%25A1szl%25C3%25B3_Lov%25C3%25A1sz/X0VB38Ee">Lovász, L.</a> (1982). <a href="http://infoscience.epfl.ch/record/164484/files/nscan4.PDF">"Factoring polynomials with rational coefficients"</a> (PDF). <a href="/facts/Mathematische_Annalen/GPhGaonc">Mathematische Annalen</a>. 261 (4): 515–534. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01457454">10.1007/BF01457454</a>. <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/1887%2F3810">1887/3810</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0682664">0682664</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:5701340">5701340</a>.</li>
<li><a href="/facts/L%25C3%25A1szl%25C3%25B3_Lov%25C3%25A1sz/X0VB38Ee">Lovász, L.</a>: An Algorithmic Theory of Numbers, Graphs, and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986</li>
<li>Malyshev, A.V. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Geometry_of_numbers">"Geometry of numbers"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li><a href="/facts/Hermann_Minkowski/tlKrpw69">Minkowski, Hermann</a> (1910), <a href="https://archive.org/details/geometriederzahl00minkrich">Geometrie der Zahlen</a>, Leipzig and Berlin: R. G. Teubner, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:41.0239.03">41.0239.03</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0249269">0249269</a>, retrieved 2016-02-28</li>
<li><a href="/facts/Wolfgang_M._Schmidt/5uFcNFt5">Wolfgang M. Schmidt</a>. Diophantine approximation. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])</li>
<li><a href="/facts/Wolfgang_M._Schmidt/5uFcNFt5">Schmidt, Wolfgang M.</a> (1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. Vol. 1467 (2nd ed.). <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-54058-X. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0754.11020">0754.11020</a>.</li>
<li><a href="/facts/Carl_Ludwig_Siegel/I91aMunI">Siegel, Carl Ludwig</a> (1989). <a href="https://archive.org/details/lecturesongeomet0000sieg">Lectures on the Geometry of Numbers</a>. <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>.</li>
<li>Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, 1993.</li>
<li>Anthony C. Thompson, Minkowski geometry, Cambridge University Press, Cambridge, 1996.</li>
<li><a href="/facts/Hermann_Weyl/4C5HhyGU">Hermann Weyl</a>. Theory of reduction for arithmetical equivalence . Trans. Amer. Math. Soc. 48 (1940) 126–164. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0002-9947-1940-0002345-2">10.1090/S0002-9947-1940-0002345-2</a></li>
<li>Hermann Weyl. Theory of reduction for arithmetical equivalence. II . Trans. Amer. Math. Soc. 51 (1942) 203–231. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1989946">10.2307/1989946</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">MSC classification, 2010, available at http://www.ams.org/msc/msc2010.html, Classification 11HXX. <a href="http://www.ams.org/msc/msc2010.html" target="_blank">http://www.ams.org/msc/msc2010.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Minkowski, Hermann (2013-08-27). Space and Time: Minkowski's papers on relativity. Minkowski Institute Press. ISBN 978-0-9879871-1-2. <a href="978-0-9879871-1-2" target="_blank">978-0-9879871-1-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Schmidt's books. Grötschel, Martin; Lovász, László; Schrijver, Alexander (1993), Geometric algorithms and combinatorial optimization, Algorithms and Combinatorics, vol. 2 (2nd ed.), Springer-Verlag, Berlin, doi:10.1007/978-3-642-78240-4, ISBN 978-3-642-78242-8, MR 1261419 <a href="978-3-642-78242-8" target="_blank">978-3-642-78242-8</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Cassels (1971) p. 203 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Grötschel et al., Lovász et al., Lovász, and Beck and Robins. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">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. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">For Kolmogorov's normability theorem, see Walter Rudin's Functional Analysis. For more results, see Schneider, and Thompson and see Kalton et al. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Kalton et al. Gardner <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Geometry of numbers open-in-new

Geometry of numbers