# Minkowski's theorem

**Minkowski's theorem** (the convex body theorem, also called Minkowski's first theorem) is a result in the geometry of numbers stating that any convex set in \( \mathbb{R}^n \) that is symmetric about the origin and has volume greater than \( 2^n \) times the determinant of a lattice must contain a lattice point other than zero. [Hermann Minkowski](https://www.edgepedia.org/hermann-minkowski) established it in 1896, and it is regarded as the foundation of the geometry of numbers as a separate division of number theory.<sup>[1](https://www.lattices.io/guides/minkowski-convex-body-theorem)</sup><sup> • </sup><sup>[2](https://encyclopediaofmath.org/wiki/Minkowski_theorem)</sup>

| Key fact | Detail |
|---|---|
| Statement | If \( K \subset \mathbb{R}^n \) is convex, centrally symmetric (\( x \in K \Rightarrow -x \in K \)), and \( \mathrm{vol}(K) > 2^n \det(\Lambda) \), then \( K \) contains a nonzero point of the lattice \( \Lambda \)<sup>[1](https://www.lattices.io/guides/minkowski-convex-body-theorem)</sup> |
| Compact case | For compact \( K \) the inequality can be relaxed to \( \mathrm{vol}(K) \ge 2^n \det(\Lambda) \)<sup>[1](https://www.lattices.io/guides/minkowski-convex-body-theorem)</sup> |
| Sharpness | The constant \( 2^n \) cannot be reduced: the open cube \( (-1,1)^n \) has volume \( 2^n \) and contains no nonzero point of \( \mathbb{Z}^n \)<sup>[1](https://www.lattices.io/guides/minkowski-convex-body-theorem)</sup> |
| Short-vector bound | Every lattice satisfies \( \lambda_1(\Lambda) \le 2\left(\det(\Lambda)/V_n\right)^{1/n} \le \sqrt{n}\,\det(\Lambda)^{1/n} \), where \( V_n \) is the volume of the unit ball<sup>[1](https://www.lattices.io/guides/minkowski-convex-body-theorem)</sup> |
| Linear forms corollary | The system \( \lvert \sum a_{1j}x_j \rvert \le c_1 \), \( \lvert \sum a_{ij}x_j \rvert < c_i \) for \( i = 2,\ldots,n \) has a nonzero integer solution if \( c_1 \cdots c_n \ge \lvert \det a_{ij} \rvert \)<sup>[2](https://encyclopediaofmath.org/wiki/Minkowski_theorem)</sup> |
| Number theory application | The Minkowski bound \( M_K = (4/\pi)^s (n!/n^n)\sqrt{\lvert d_K \rvert} \) implies every ideal class of a number field has an integral ideal of norm at most \( M_K \), so the class group is finite<sup>[3](https://wstein.org/books/ant/ant/ch_classgroup.html)</sup> |

## Statement

Let \( \Lambda \subset \mathbb{R}^n \) be a full-rank lattice with determinant \( \det(\Lambda) \), the volume of one fundamental cell. The theorem says: if \( K \) is convex and centrally symmetric about the origin, and

\[ \mathrm{vol}(K) > 2^n \det(\Lambda), \]

then \( K \) contains a nonzero vector of \( \Lambda \).<sup>[1](https://www.lattices.io/guides/minkowski-convex-body-theorem)</sup> In the formulation of the Encyclopedia of Mathematics, for a closed convex body \( N \) symmetric with respect to the origin with volume \( V(N) \), every point lattice \( \Lambda \) of determinant \( d(\Lambda) \) with \( V(N) \ge 2^n d(\Lambda) \) has a point in \( N \) distinct from \( 0 \).<sup>[2](https://encyclopediaofmath.org/wiki/Minkowski_theorem)</sup>

Because \( K \) is symmetric, lattice points come in pairs: if \( v \) is in \( K \), so is \( -v \). The theorem guarantees one such pair; it says nothing about how many lattice points there are or where they sit inside \( K \).<sup>[1](https://www.lattices.io/guides/minkowski-convex-body-theorem)</sup> An equivalent formulation uses the critical determinant \( \Delta(N) \) of the body: \( \Delta(N) \ge 2^{-n} d(\Lambda) \).<sup>[2](https://encyclopediaofmath.org/wiki/Minkowski_theorem)</sup>

## Proof idea

The proof is a pigeonhole argument, known as Blichfeldt's lemma, applied to the scaled set \( \tfrac{1}{2}K \).<sup>[1](https://www.lattices.io/guides/minkowski-convex-body-theorem)</sup> Blichfeldt's principle states that a bounded region \( D \subset \mathbb{R}^n \) with \( v(D) > 1 \) contains two distinct vectors whose difference is an integer vector.<sup>[4](https://www.math.uzh.ch/gorodnik/nt/lecture14.pdf)</sup>

The argument runs as follows. Set \( Y = \tfrac{1}{2}X \); then \( \mathrm{vol}(Y) = 2^{-n}\,\mathrm{vol}(X) > \mathrm{Vol}(\Lambda) \). If the translates \( v + Y \) for \( v \in \Lambda \) were pairwise disjoint, their total volume could not exceed the volume of a fundamental domain, a contradiction. So two distinct lattice vectors \( v, w \) satisfy \( (v+Y) \cap (w+Y) \ne \emptyset \), giving \( v - w \in X \).<sup>[5](https://www.math.ucla.edu/~sharifi/notes/algnum-ch04.html)</sup> The symmetry and convexity hypotheses enter exactly at this step: symmetry puts \( -2y \in X \) whenever \( 2y \in X \), and convexity then places the midpoint \( \tfrac{1}{2}(2y') + \tfrac{1}{2}(-2y) \) in \( X \), turning a difference of two points of \( \tfrac{1}{2}K \) into a point of \( K \).<sup>[1](https://www.lattices.io/guides/minkowski-convex-body-theorem)</sup>

The constant is genuinely sharp. The open cube \( (-1,1)^n \) has volume exactly \( 2^n \) and contains no nonzero integer point, so no smaller constant can work for all convex symmetric bodies.<sup>[1](https://www.lattices.io/guides/minkowski-convex-body-theorem)</sup>

## The short-vector corollary

Applying the theorem to a cube and to a ball yields a bound on the shortest nonzero lattice vector. The cube \( [-r, r]^n \) has volume \( (2r)^n \), which equals \( 2^n \det(\Lambda) \) when \( r = \det(\Lambda)^{1/n} \); the compact version gives a nonzero lattice vector with every coordinate at most \( r \) in absolute value, hence length at most \( \sqrt{n}\,r \). A ball of radius \( r \) gives the sharper bound

\[ \lambda_1(\Lambda) \le 2\left(\frac{\det(\Lambda)}{V_n}\right)^{1/n} \le \sqrt{n}\,\det(\Lambda)^{1/n}, \]

where \( V_n \) is the volume of the unit ball. Squaring the ball bound and dividing by \( \det(\Lambda)^{2/n} \) gives the upper bound \( \gamma_n \le 4/V_n^{2/n} \) on Hermite's constant, which grows roughly like \( 2n/(\pi e) \) for large \( n \). For \( n = 2 \) it gives \( 4/\pi \approx 1.273 \), while the true value is \( 2/\sqrt{3} \approx 1.155 \).<sup>[1](https://www.lattices.io/guides/minkowski-convex-body-theorem)</sup> These are existence results: finding such a vector is the shortest vector problem, which is hard in high dimension, and the LLL algorithm reaches only an exponential approximation of it.<sup>[1](https://www.lattices.io/guides/minkowski-convex-body-theorem)</sup> The bound also cannot be asymptotically improved as a function of the determinant: there is a constant \( c \) such that for every dimension \( n \) some \( n \)-dimensional lattice has \( \lambda_1 > c\sqrt{n}\,\det(\Lambda)^{1/n} \).<sup>[6](https://cseweb.ucsd.edu/classes/fa25/cse206A-a/Lec2-Minkowski.pdf)</sup>

## Applications in number theory

**Diophantine approximation.** Minkowski's theorem proves Dirichlet's theorem on simultaneous approximation: for real numbers \( \theta_1, \ldots, \theta_n \) and any integer \( Q > 1 \), there exist integers \( p_1, \ldots, p_n \) and \( q \) with \( 1 \le q \le Q \) such that \( \lvert \theta_i - p_i/q \rvert < 1/(qQ^{1/n}) \) for all \( i \). The proof applies the theorem to a box in \( \mathbb{R}^{n+1} \) of volume \( 2(Q+1)(2Q^{-1/n})^n > 2^{n+1} \).<sup>[4](https://www.math.uzh.ch/gorodnik/nt/lecture14.pdf)</sup> The theorem on linear forms, a corollary of the convex body theorem, was published by Minkowski in the same 1896 work; the question of when its first inequality can be made strict was solved by G. Hajós.<sup>[2](https://encyclopediaofmath.org/wiki/Minkowski_theorem)</sup>

**Sums of two squares.** The theorem also yields the classical characterization that a positive integer is a sum of two squares exactly when, in its prime factorization, every prime congruent to 3 modulo 4 appears to an even power.<sup>[4](https://www.math.uzh.ch/gorodnik/nt/lecture14.pdf)</sup>

**Class groups of number fields.** The most prominent application is the finiteness of the [ideal class group](https://www.edgepedia.org/ideal-class-group). The Minkowski bound

\[ M_K = \left(\frac{4}{\pi}\right)^{s} \frac{n!}{n^{n}} \cdot \sqrt{\lvert d_K \rvert} \]

where \( n \) is the degree of the number field \( K \), \( s \) the number of pairs of complex conjugate embeddings, and \( d_K \) its discriminant, guarantees that every ideal class contains an integral ideal of norm at most \( M_K \). Since only finitely many ideals have norm below any given integer, the class group is finite.<sup>[3](https://wstein.org/books/ant/ant/ch_classgroup.html)</sup> The same reasoning gives Minkowski's lower discriminant bound, from which \( \lvert \Delta_F \rvert > 1 \) for every number field \( F \ne \mathbb{Q} \).<sup>[7](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/mollin)</sup> The bound also allows actual computation of class groups in small cases; for \( K = \mathbb{Q}(\sqrt{10}) \) it equals \( \sqrt{10} \approx 3.162 \), and checking ideals of norm 1, 2, and 3 shows the class group has order 2.<sup>[3](https://wstein.org/books/ant/ant/ch_classgroup.html)</sup>

## History

Minkowski initiated the geometry of numbers in 1890, and the first 240 pages of his *Geometrie der Zahlen* appeared in 1896, with the full 256-page book published in 1910.<sup>[8](https://mathshistory.st-andrews.ac.uk/Biographies/Minkowski/)</sup> His 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.<sup>[8](https://mathshistory.st-andrews.ac.uk/Biographies/Minkowski/)</sup> Work on the geometry of numbers led on to convex bodies and packing problems.<sup>[8](https://mathshistory.st-andrews.ac.uk/Biographies/Minkowski/)</sup>

## Related results and refinements

A generalization of Minkowski's theorem to non-convex bodies is Blichfeldt's theorem; together, the theorems of Minkowski and Blichfeldt allow upper estimates of the arithmetic minima of distance functions.<sup>[2](https://encyclopediaofmath.org/wiki/Minkowski_theorem)</sup> A refinement employing [Fourier series](https://www.edgepedia.org/fourier-series) was given by C. L. Siegel, and a different refinement is Minkowski's theorem on successive minima; both have applications in algebraic number theory and Diophantine approximation.<sup>[2](https://encyclopediaofmath.org/wiki/Minkowski_theorem)</sup> Minkowski's second convex body theorem, bounding the successive minima of a lattice, remains a working tool in modern research, for example in parametric geometry of numbers for simultaneous Diophantine approximation.<sup>[9](https://arxiv.org/abs/2010.01000)</sup>

The theorem also reaches into computation. Ravi Kannan, a computer scientist then at Carnegie-Mellon University, used Minkowski's convex body theorem and related geometry-of-numbers results in a 1987 algorithm for integer programming whose running time depends on the number \( n \) of variables as \( n^{O(n)} \), improving the previously best per-variable factor from exponential in \( n \) to \( O(n^{5/2}) \).<sup>[10](https://pubsonline.informs.org/doi/10.1287/moor.12.3.415)</sup>

## References

1. [Minkowski's Convex Body Theorem: Statement and Proof, Lattices.io](https://www.lattices.io/guides/minkowski-convex-body-theorem)
2. [Minkowski theorem, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Minkowski_theorem)
3. [Chapter 7: Finiteness of the Class Group, William Stein, Algebraic Number Theory, a Computational Approach](https://wstein.org/books/ant/ant/ch_classgroup.html)
4. [Lecture 14: Geometry of Numbers, O. Gorodnik, University of Zurich](https://www.math.uzh.ch/gorodnik/nt/lecture14.pdf)
5. [Algebraic Number Theory, Chapter 4: Geometry of Numbers, Romyar Sharifi, UCLA](https://www.math.ucla.edu/~sharifi/notes/algnum-ch04.html)
6. [Lecture 2: Minkowski, CSE 206A, UC San Diego](https://cseweb.ucsd.edu/classes/fa25/cse206A-a/Lec2-Minkowski.pdf)
7. [Discrete Mathematics and Its Applications (Mollin), University of Edinburgh archive](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/mollin)
8. [Hermann Minkowski (1864–1909), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Minkowski/)
9. [Going-up theorems for simultaneous Diophantine approximation, arXiv:2010.01000](https://arxiv.org/abs/2010.01000)
10. [Ravi Kannan (1987). Minkowski's Convex Body Theorem and Integer Programming. Mathematics of Operations Research 12(3):415–440.](https://pubsonline.informs.org/doi/10.1287/moor.12.3.415)

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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*

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