\section{Minkowski's Theorem}

\theorem{Blichfeldt's Theorem}
Let $X$ be a finite connected region. If the volume of $X$ is greater than $1$, $X$ must contain two distinct points that differ by an element of $\mathbb{Z}^n$. In other words, there exist distinct $x, y \in X$ so that $x - y \in \mathbb{Z}^n$.

\vspace{2mm}

Intuitively, this means that you can translate $X$ to cover two lattice points at the same time.


\problem{}
Draw a region in $\mathbb{R}^2$ with volume greater than 1 that contains no lattice points. Find two points in that region which differ by an integer vector.
\hint{Area is two-dimensional volume.}

\vfill


\problem{}
The following picture gives an idea for the proof of Blichfeldt's theorem in $\mathbb{Z}^2$. Explain the picture and complete the proof.

\begin{center}
	\includegraphics[angle=90,width=0.5\linewidth]{proof.png}
\end{center}

\begin{solution}
	The fundamental region of $\mathbb{Z}^2$ tiles the plane. Translate these tiles by lattice vectors to stack them on the fundamental region. Then since the union of the intersections of X with these tiles has area greater 1 and they are stacked on a region of area 1, there must be an overlap by a generalization of the pigeonhole principle (if there were no overlap then the sum of the areas would be less than or equal to 1). Take points $x, y$ in the overlap. Then $x - y$ is a lattice point corresponding to the difference in translates, which were lattice points. Hence, $x - y \in \mathbb{Z}^2$.
\end{solution}

\vfill
\pagebreak


%\problem{}
%Does your proof of Blichfeldt's theorem in $\mathbb{Z}^2$ extend to a proof of Blichfeldt's theorem in $\mathbb{Z}^n$?
%\vfill


\problem{}
Let $X$ be a region $\in \mathbb{R}^2$ of volume $k$. How many integral points must $X$ contain after a translation?

\begin{solution}
	$\lceil k \rceil$
\end{solution}

\vfill

\definition{}
A region $X$ is \textit{convex} if the line segment connecting any two points in $X$ lies entirely in $X$.

\problem{}
\begin{itemize}
	\item Draw a convex region in the plane.
	\item Draw a region that is not convex.
\end{itemize}

\vfill
\pagebreak



\definition{}
We say a region $X$ is \textit{symmetric} if for all points $x \in X$, $-x$ is also in $X$.

\problem{}
\begin{itemize}
	\item Draw a symmetric region.
	\item Draw an asymmetric region.
\end{itemize}

\vfill

\theorem{Minkowski's Theorem}<mink>
Every convex set in $\mathbb{R}^n$ that is symmetric with respect to the origin and which has a volume greater than $2^n$ contains an integral point that isn't zero.


\problem{}
Draw a few sets that satisfy \ref{mink} in $\mathbb{R}^2$. \par
What is the simplest region that has the properties listed above?

\vfill


\problem{}
Let $K$ be a region in $\mathbb{R}^2$ satisfying \ref{mink}. \par
Let $K'$ be this region scaled by $\frac{1}{2}$.

\begin{itemize}
	\item How does the volume of $K'$ compare to $K$?
	\item Show that the sum of any two points in $K'$ lies in $K$ \hint{Use convexity.}
	\item Apply Blichfeldt's theorem to $K'$ to prove Minkowski's theorem in $\mathbb{R}^2$.
\end{itemize}

\vfill

\problem{}
Let $K$ be a region in $\mathbb{R}^n$ satisfying \ref{mink}. Scale this region by $\frac{1}{2}$, called $K' = \frac{1}{2}K$.

\begin{itemize}
	\item How does the volume of $K'$ compare to $K$?
	\item Show that the sum of any two points in $K'$ lies in $K$
	\item Apply Blichfeldt's theorem to $K'$ to prove Minkowski's theorem.
\end{itemize}

\begin{solution}
	\begin{itemize}
		\item The volume of $K'$ is $\frac{1}{2^n}$ the volume of $K$.
		\item Take $x, y \in K'$. It follows that $2x, 2y \in K$. Since $K$ is convex, we have that the midpoint of the line segment between $2x$ and $2y$ is in $K$, and so $\frac{2x + 2y}{2} = x + y \in K$.
		\item Since the volume of $K$ is greater than $2^n$, we have the volume of $K'$ is greater than one.
		Applying Blichfeldt's theorem, we can find two distinct points $x, y \in K'$ such that $x - y \in \mathbb{Z}^n$. Since $K'$ is symmetric with respect to the origin, we have that $-y \in K'$. Therefore, $x + (-y) \in K$ by the previous part. $x \neq y, x - y \neq 0$, so we have found a nontrivial integer point in $K$.

	\end{itemize}
\end{solution}

\vfill
\pagebreak