# Geometry of numbers
**Geometry of numbers** is a branch of number theory that proves arithmetic statements by geometric means, studying how lattice points (points with integer coordinates, or points of a general lattice) interact with convex bodies in [Euclidean space](https://www.edgepedia.org/euclidean-space). It was founded by [Hermann Minkowski](https://www.edgepedia.org/hermann-minkowski), who initiated the subject in 1890 and whose monograph *Geometrie der Zahlen* appeared in 1910, a year after his death, with the first 240 of its 256 pages already published in 1896.[1](https://mathshistory.st-andrews.ac.uk/Biographies/Minkowski/) The field's central result, [Minkowski's theorem](https://www.edgepedia.org/minkowskis-theorem), gives a criterion purely in terms of volume for a symmetric convex set to contain a nonzero lattice point.[2](https://www.math.cmu.edu/~ttkocz/teaching/1920/conv-discr-geom-notes.pdf)
| Key fact | Detail |
|---|---|
| Founder | Hermann Minkowski (1864–1909); subject initiated in 1890, monograph *Geometrie der Zahlen* published 1910[1](https://mathshistory.st-andrews.ac.uk/Biographies/Minkowski/) |
| First theorem | A convex set in \( \mathbb{R}^d \), symmetric about the origin, with volume greater than \( 2^d \), contains a nonzero point of \( \mathbb{Z}^d \)[2](https://www.math.cmu.edu/~ttkocz/teaching/1920/conv-discr-geom-notes.pdf) |
| Sharpness | The open cube \( (-1,1)^d \) has volume \( 2^d \) and no nonzero lattice point, so the constant \( 2^d \) cannot be lowered[2](https://www.math.cmu.edu/~ttkocz/teaching/1920/conv-discr-geom-notes.pdf) |
| Second theorem | The successive minima \( \lambda_1 \le \cdots \le \lambda_n \) of a symmetric convex body satisfy \( \frac{2^n}{n!} \cdot \frac{\det(\Lambda)}{V_n} \le \prod_{k=1}^n \lambda_k \le 2^n \cdot \frac{\det(\Lambda)}{V_n} \)[3](https://www.lattices.io/intro/geometry-of-numbers) |
| Number-theoretic payoff | The Minkowski bound shows every ideal class of a number field contains an integral ideal of bounded norm, proving the ideal class group is finite[4](https://math.mit.edu/classes/18.785/2016fa/LectureNotes14.pdf) |
| Algorithmic reach | Lenstra's algorithm solves integer linear programming in polynomial time when the dimension is fixed, using geometry-of-numbers tools[5](https://www.epfl.ch/labs/disopt/teaching/page-31585-en-html/page-10548-en-html/) |
| Modern cryptography | Lattice problems (SVP, CVP) underlie the NIST post-quantum standards ML-KEM (FIPS 203) and ML-DSA (FIPS 204), approved August 2024[6](https://csrc.nist.gov/News/2024/postquantum-cryptography-fips-approved) |
## Lattices and convex bodies
The subject is concerned with the interplay of compact convex 0-symmetric sets and lattices in Euclidean spaces.[7](https://www1.claremontmckenna.edu/pages/faculty/lenny/papers/GNT_lecture_notes.pdf) A set \( K \subset \mathbb{R}^n \) is convex if the line segment between any two of its points lies in \( K \), and it is 0-symmetric (symmetric about the origin) if \( -x \in K \) whenever \( x \in K \).[7](https://www1.claremontmckenna.edu/pages/faculty/lenny/papers/GNT_lecture_notes.pdf) A lattice in an \( n \)-dimensional Euclidean space \( V \) is a subgroup of the form \( L = \mathbb{Z} \cdot x_1 + \cdots + \mathbb{Z} \cdot x_k \) with linearly independent generators; the lattice is complete, or of maximal rank, when \( k = n \).[8](https://websites.math.leidenuniv.nl/algebra/ant.pdf) For a complete lattice, the co-volume \( \mathrm{vol}(V/L) \) is the volume of the parallelepiped spanned by a basis, which serves as a fundamental domain: every point of \( V \) has a unique representation as a point of the domain plus a lattice vector.[8](https://websites.math.leidenuniv.nl/algebra/ant.pdf)
**Why convexity and symmetry are essential.** A volume criterion for lattice points is possible only for sets with both properties. A rectangle in the plane with \( x \)-coordinates between \( 1/3 \) and \( 2/3 \) and \( y \)-coordinates between \( -t \) and \( t \) is convex for every \( t \), has area \( 2t/3 \) that grows arbitrarily large, yet contains no integer points at all; it fails symmetry. Conversely, the union of a half-plane and its negative is 0-symmetric but not convex, and can likewise have arbitrarily large area with no integer points.[7](https://www1.claremontmckenna.edu/pages/faculty/lenny/papers/GNT_lecture_notes.pdf)
## Minkowski's first theorem
Minkowski's theorem states that every symmetric convex set which is large enough in terms of its volume contains a nonzero lattice point.[2](https://www.math.cmu.edu/~ttkocz/teaching/1920/conv-discr-geom-notes.pdf) In the general lattice form: if \( L \) is a complete lattice in an \( n \)-dimensional Euclidean space \( V \) and \( X \subset V \) is bounded, convex, and symmetric with
\[ \mathrm{vol}(X) > 2^n \cdot \mathrm{vol}(V/L), \]
then \( X \) contains a nonzero point of \( L \).[8](https://websites.math.leidenuniv.nl/algebra/ant.pdf) For the integer lattice \( \mathbb{Z}^d \), whose fundamental domain has volume 1, the threshold is \( 2^d \).[2](https://www.math.cmu.edu/~ttkocz/teaching/1920/conv-discr-geom-notes.pdf)
The proof idea is a packing argument. Halve the body to \( Y = \frac{1}{2}X \), which has volume \( 2^{-n} \mathrm{vol}(X) \), exceeding the co-volume of the lattice. If the translates \( v + Y \) for lattice points \( v \) were pairwise disjoint, their total volume could not exceed that of the fundamental domain; so two translates overlap, and the difference of a point in each lies in \( X \) by convexity and symmetry, giving a nonzero lattice vector in \( X \).[9](https://www.math.ucla.edu/~sharifi/notes/algnum-ch04.html)
The constant is sharp, and the hypotheses matter at the boundary. The open cube \( (-1,1)^d \) has volume exactly \( 2^d \) and contains no nonzero lattice point, so strict inequality cannot be relaxed in general.[2](https://www.math.cmu.edu/~ttkocz/teaching/1920/conv-discr-geom-notes.pdf) If \( K \) is additionally compact, the conclusion does hold at volume \( 2^d \): one applies the theorem to the slightly enlarged bodies \( (1 + 1/n)K \) and extracts a fixed nonzero lattice point by compactness.[2](https://www.math.cmu.edu/~ttkocz/teaching/1920/conv-discr-geom-notes.pdf) Volume alone also cannot promise *which* lattice point appears: for any primitive lattice vector \( x \), there is a symmetric convex set of volume \( 2^d \) whose only nonzero lattice points are \( \pm x \).[2](https://www.math.cmu.edu/~ttkocz/teaching/1920/conv-discr-geom-notes.pdf)
## Successive minima and the second theorem
For a symmetric convex body \( K \) and lattice \( \Lambda \), the successive minima \( \lambda_1 \le \lambda_2 \le \cdots \le \lambda_n \) measure how far \( K \) must be scaled to capture linearly independent lattice points: \( \lambda_1 \) is the smallest magnification that captures any nonzero lattice point, and each subsequent \( \lambda_k \) the smallest that captures a new direction independent of the previous ones.[3](https://www.lattices.io/intro/geometry-of-numbers) • [10](https://mathweb.ucsd.edu/~jverstra/minkowski.pdf)
**Minkowski's second theorem** bounds the product of these minima on both sides:
\[ \frac{2^n}{n!} \cdot \frac{\det(\Lambda)}{V_n} \le \prod_{k=1}^n \lambda_k(\Lambda) \le 2^n \cdot \frac{\det(\Lambda)}{V_n}, \]
where \( V_n = \pi^{n/2}/\Gamma(n/2+1) \) is the volume of the Euclidean unit ball in \( \mathbb{R}^n \). The upper bound says the successive minima cannot all be large in a dense lattice; the lower bound says they cannot all be small, a constraint on how well-packed any lattice can be.[3](https://www.lattices.io/intro/geometry-of-numbers) Minkowski also proved the sharper upper bound \( \prod_{k=1}^n \lambda_k \le \gamma_n^{n/2} \cdot \det(\Lambda) \), where \( \gamma_n \) is Hermite's constant.[3](https://www.lattices.io/intro/geometry-of-numbers) Unlike the first theorem, computing the successive minima is hard: finding \( \lambda_2, \ldots, \lambda_n \) is at least as hard as the [shortest vector problem](https://www.edgepedia.org/shortest-vector-problem).[3](https://www.lattices.io/intro/geometry-of-numbers) Minkowski's own proof of the second theorem was lengthy and was criticized as obscure; later work has given shorter proofs along his lines and bounds on lattice point counts in terms of the successive minima.[11](https://arxiv.org/abs/math/0204158) The successive minima have since become a standard tool in convex and discrete geometry, connected to the lattice point enumerator, intrinsic volumes, and Siegel's lemma.[12](https://arxiv.org/abs/2304.00120)
## Applications in algebraic number theory
The classical finiteness theorems of [algebraic number theory](https://www.edgepedia.org/algebraic-number-theory) rest on Minkowski's theorem. A number ring embeds naturally as a lattice in a finite-dimensional real vector space, and this is the special fact behind proofs that the [Picard group](https://www.edgepedia.org/picard-group) of a number ring is finite and that its unit group is in many cases finitely generated.[8](https://websites.math.leidenuniv.nl/algebra/ant.pdf)
**Finiteness of the class group.** The ideal class group of a number field \( K \) is the quotient of the group of fractional ideals by the principal ones; its order is the class number \( h_K \). Minkowski's theory gives a bound: every ideal class contains an integral ideal \( I \subseteq \mathcal{O}_K \) with norm at most the Minkowski constant \( m_K \), and the set of integral ideals with norm below any fixed bound is finite, so the class group is finite.[4](https://math.mit.edu/classes/18.785/2016fa/LectureNotes14.pdf) In one common form, every ideal class contains an ideal with
\[ N(I) \le \left(\frac{2}{\pi}\right)^{s} |\Delta_K|^{1/2}, \]
where \( s \) is the number of complex places and \( \Delta_K \) the discriminant.[7](https://www1.claremontmckenna.edu/pages/faculty/lenny/papers/GNT_lecture_notes.pdf) The finiteness of the class number was first proved by Minkowski using his geometry of numbers.[7](https://www1.claremontmckenna.edu/pages/faculty/lenny/papers/GNT_lecture_notes.pdf) The bound is effective: to determine the class group one factors all primes up to the bound, and the Minkowski bound allows actual computation of class numbers and sometimes of the full class group.[9](https://math.ucla.edu/~sharifi/notes/algnum-ch04.html)
The same circle of ideas yields a lower bound for the discriminant, \( |\mathrm{disc}(\mathcal{O}_K)| \ge \left(\frac{n^n}{n!}\right)^2 \left(\frac{\pi}{4}\right)^{2s} \), which grows exponentially with the degree \( n \); a consequence is that \( \mathbb{Q} \) has no nontrivial [unramified extensions](https://www.edgepedia.org/unramified-extension).[4](https://math.mit.edu/classes/18.785/2016fa/LectureNotes14.pdf) Minkowski's 1907 book *Diophantische Approximationen* gave an elementary account of the geometry of numbers and its applications to [Diophantine approximation](https://www.edgepedia.org/diophantine-approximation) and algebraic numbers.[1](https://mathshistory.st-andrews.ac.uk/Biographies/Minkowski/)
## Algorithmic directions and cryptography
The traditional geometry of numbers asks under what conditions a convex set contains a lattice point; a parallel algorithmic line asks how to find such a point when one exists.[5](https://www.epfl.ch/labs/disopt/teaching/page-31585-en-html/page-10548-en-html/) Integer linear programming, the problem of finding a lattice point in a polyhedron, is NP-complete in general but solvable in polynomial time when the dimension is fixed, a result of [Hendrik Lenstra](https://www.edgepedia.org/hendrik-lenstra) that exploited geometry-of-numbers tools and stimulated deep research on algorithms in this area, including lattice basis reduction, the [LLL algorithm](https://www.edgepedia.org/lenstra-lenstra-lovasz-lattice-basis-reduction-algorithm), and Barvinok's algorithm for counting integer points.[5](https://www.epfl.ch/labs/disopt/teaching/page-31585-en-html/page-10548-en-html/)
The hardness side of lattice theory now anchors post-quantum cryptography. The security of lattice-based cryptosystems relies on the computational difficulty of the shortest vector problem (SVP) and the closest vector problem (CVP) and their generalizations; SVP is a ball packing problem and CVP a ball covering problem.[13](https://arxiv.org/abs/2506.23438) In August 2024, NIST approved the Federal Information Processing Standards FIPS 203 (ML-KEM, a module-lattice key-encapsulation mechanism derived from [CRYSTALS-Kyber](https://www.edgepedia.org/crystals-kyber)), FIPS 204 (ML-DSA, a module-lattice digital signature standard derived from CRYSTALS-Dilithium), and [FIPS 205](https://www.edgepedia.org/fips-205) (a hash-based signature standard), designed to resist attacks by quantum computers.[6](https://csrc.nist.gov/News/2024/postquantum-cryptography-fips-approved)
## References
1. [Hermann Minkowski (1864–1909), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Minkowski/)
2. [T. Tkocz, Minkowski's theorem in geometry of numbers, Carnegie Mellon University lecture notes](https://www.math.cmu.edu/~ttkocz/teaching/1920/conv-discr-geom-notes.pdf)
3. [Geometry of Numbers: Minkowski's Theorems Explained, Lattices.io](https://www.lattices.io/intro/geometry-of-numbers)
4. [18.785 Number Theory I, Lecture 14, MIT](https://math.mit.edu/classes/18.785/2016fa/LectureNotes14.pdf)
5. [Integer Points in Polyhedra, DISOPT, EPFL](https://www.epfl.ch/labs/disopt/teaching/page-31585-en-html/page-10548-en-html/)
6. [Announcing Approval of Three FIPS for Post-Quantum Cryptography, NIST CSRC](https://csrc.nist.gov/News/2024/postquantum-cryptography-fips-approved)
7. [Lenny Taelman (ed. notes), Geometric Number Theory, Claremont McKenna College](https://www1.claremontmckenna.edu/pages/faculty/lenny/papers/GNT_lecture_notes.pdf)
8. [Algebraic Number Theory, Chapter 5: Geometry of numbers, Leiden University](https://websites.math.leidenuniv.nl/algebra/ant.pdf)
9. [Romyar Sharifi, Algebraic Number Theory, Chapter 4: Geometry of numbers, UCLA](https://www.math.ucla.edu/~sharifi/notes/algnum-ch04.html)
10. [J. H. Verstraete, Notes on geometry of numbers, UC San Diego](https://mathweb.ucsd.edu/~jverstra/minkowski.pdf)
11. [Successive Minima and Lattice Points, arXiv:math/0204158](https://arxiv.org/abs/math/0204158)
12. [I. Aliev, M. Henk, Minkowski's successive minima in convex and discrete geometry, arXiv:2304.00120](https://arxiv.org/abs/2304.00120)
13. [C. Zong, Some Mathematical Problems Behind Lattice-Based Cryptography, arXiv:2506.23438](https://arxiv.org/abs/2506.23438)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Diophantine problems and approximation › Metric Diophantine approximation and geometry of numbers*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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