** Minkowski's Convex Body Theorem**

Suppose $S$ is a convex set in an "n-dimensional" vector space in a lattice $**L**$ which is **symmetric** *about the origin.* Then, Minkowski's Theorem states that if the volume of the set **$V(S) > 2^nd(L)$**, then $S$ must contain at least one lattice point other than the origin.

Before diving deeper, let's first clearly understand some terms used in the theorem..

**Convex**

A set $C$ is said to be convex in $\mathbb{R}^n$ if for all points $a,b \in \mathbb{C}$, the segment joining the points $a$ and $b$ is completely contained in $C$. \newline

An example for a set in $\mathbb{R^2}$ is as follows.

**Symmetric about the origin**

A set $C$ is said to be symmetric about the origin if for every point $x \in C$, $-x \in C$. In other words, the reflection of every point in $C$ across the origin must also be in $C$. For example

**Lattice**

**Determinant of a lattice**

**Blichfeldt's Theorem**

*If R is a bounded set in $\mathbb{R}^2$ with area greater than 1, then R contains two distinct points $(x_1,y_1)$ and $(x_2,y_2)$ such that the point*

**${(x_2-x_1,y_2-y_1)}$**is an integer point in $\mathbb{R}^2$*Proof:*Let $S = \{(x,y) | 0 \le x < 1 \text{and} 0 \le y <1\}$. For every point $a \in \mathbb{Z}^2$, define $S_a = S + a$ be the translation of $S$ along the line segment with endpoints $(0,0)$ and $a$. Note that $a$ will be the only integer point in $S_a$ Since, $R$ is bounded, we know that there will be a finite number of points $z$ such that $R_z = S_z \cap R$ is non empty. Let $R_z-z$ be the translation of $R_z$ back to $S$ along the line segment with endpoints $(0,0)$ and $z$. Since the translations are plane isometry, the area of the sets during translation will be preserved, thus, $A(R_z-z) = A(R_z)$. We have,

*Remark:*

*Note that this theorem is also true for "non-integer lattices", say $L$. The only difference would be that $R$ should have an area greater than $d(L)$. The proof will be similar to this.*

**Minkowski's Theorem**". Now, let's move on to prove a simpler version of "

**Minkowski's Convex Body Theorem**".

**Simpler version**

*Let $R$ be a convex region in $\mathbb{R}^2$ that is symmetric about the origin and has an area greater than $2^2d(\mathbb{Z}^2) = 4$. Then $R$ contains an integer point other than the origin.*

*Proof:*Define the set $R'= \left\{\frac{1}{2}x | x \in R\right\}$. Since $R'$ is just $R$ scaled down by a factor of $0.5$, it is also convex and symmetric about the origin. Moreover, $A(R') = \frac{1}{4} A(R) > 1$. By

**Blichfeldt's Theorem**, there exists two distinct points $m,n \in R'$ such that $m-n$ is an integer point. Note that $2m, 2n \in R$. Since, $R$ is symmetric about the origin, $-2n \in R$. We know that $R$ is convex. Thus, every point on the line segment with endpoints $2m$ and $-2n$ lies inside $R$. Therefore, their midpoint also lies in $R$. We have

*Note that the symmetric condition yield that along with $m-n$, the point $-(m-n)$ must also lie in $R$. Thus, there are at least 3 integer points in $R$.*

**Minkowski's Theorem for Arbitrary Lattices**

### Applications of Minkowski's Theorem

**The two squares theorem:**

*For a prime $p \equiv 1 \pmod 4$, we can always find some $a,b \in \mathbb{Z}$, such that $p = a^2 + b^2$*

*Proof:*By, Fermat's Christmas Theorem, $(-1)$ is a quadratic residue modulo $p$. Thus, there exists some $b$ such that $q^2 \equiv -1 \pmod p$. Let $z_1 = (1,q)$ and $z_2 = (0,p)$. Define a lattice $L$ such that $z_1$, $z_2$, $(0,0)$ and $(z_1-z_2)$ are all lattice points. Note that these four points form a parallelogram with area $p$. Thus, $d(L) = p$. \newline

**Akshat Pandey.**

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