We present a proof of a famous theorem of *Hermann Minkowski* which spawned a beautiful branch of number theory known as the *Geometry of Numbers*. This theorem gives a condition on the volume of a centrally symmetric convex body in -dimensional Euclidean space to contain at least one lattice point except the trivial point .

**Definitions and Notations**

**Definition 1 **In an -dimensional vector space over the field , a **lattice** is a free abelian subgroup of of the following form

where are linearly independent vectors in . Then is a free abelian group of rank . A lattice is said to be **complete** if . Assuming to be complete, we define the **fundamental mesh** in with respect the the to be

The volume of the lattice is defined to be the volume of the fundamental mesh denoted as , *i.e.* . There are a few equivalent expressions for this volume. One expression convenient for computations is given by the following relation

where is the usual inner-product (or a symmetric positive definite bilinear form) defined on the vector space .

**Definition 2 (Convex and Centrally Symmetric Body)**

A subset of is said to be **centrally symmetric** if for any point we have , and the subset is **convex** if for any two distinct points in the line segment is contained in . For example, a **ball** is convex in but a **solid torus** is not.

**Minkowski’s Lattice Point Theorem**

** Theorem:** Let be an -dimensional Euclidean vector space (-vector space) and be a complete lattice of . Let be a centrally symmetric convex subset of such that

Then contains at least one point other than in .

**Proof:** Consider the *dilation* . For any , consider the translated sets . We will show that there exist such that and

For the sake of contradiction, assume the sets are pairwise disjoint. Then the intersections are also disjoint for a fundamental mesh . This gives us the following inequality involving volumes

We know that translation preserves volumes, so~~ ~~we consider the following translations

Hence for each we have

We claim that the cover as varies over . Let . Since is a set of linearly independent vectors in and , it follows that is a basis of . Then such that . We know that every real number can be written as where and . Then

Note that and . Taking we observe that . Hence, . Since for all we have . Hence , and so the sets cover . Therefore, we finally have

which is a contradiction to our initial assumption since . Hence, we can choose such that

Therefore, there exist , such that

Since is centrally symmetric and convex, and thus Therefore and . This completes the proof of the theorem.

**Remark**

Since we also have and . This means contains at least two distinct lattice points. What else can you find hidden in the proof?

**A Useful Corollary**

Taking and , we observe that any convex centrally symmetric body in of volume strictly bigger that contains at least one point with integer coordinates other than .

**Minkowski’s Theorem in Action**

Now we present a number theoretic application of Minkowski’s theorem. The following result was proved by **Axel Thue** using the **pigeonhole principle**. We give a proof using Minkowski’s lattice point theorem.

**Theorem:*** Primes of the form can be expressed as a sum of two squares. *

**Proof.** Let be a prime of the form . Then is a quadratic residue modulo or, equivalently, there exists such that . Consider the two vectors and in . Let for some . This gives us and hence . Therefore are linearly independent. Then is a complete lattice in with .

Let . There exist such that . This implies . Hence,

Consider the open disc of radius centered at the origin . We have,

Thus, is convex and centrally symmetric. By Minkowski’s theorem, there exists a lattice point apart from the origin in . Let this point be . Then and and hence . We are done!

**References**

[1] Andreescu, T. and Dospinescu, G., 2008. Problems from the Book.

[2] Neukirch, J., 2013. Algebraic number theory (Vol. 322). Springer Science & Business Media.

[3] Cassels, J.W.S., 2012. An introduction to the geometry of numbers. Springer Science & Business Media.