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2023-05-15 10:33:03 -07:00
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Advanced/Lattices/parts/1 minkowski.tex
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@ -1,9 +1,9 @@
\section{Minkowski's Theorem}
\theorem{Blichfeldt'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 $x - y \in \mathbb{Z}^n$.
\vfill{4mm}
\vspace{2mm}
Intuitively, this means that you can translate $X$ to cover two lattice points at the same time.
@ -15,8 +15,8 @@ Draw a region in $\mathbb{R}^2$ with volume greater than 1 that contains no latt
\vfill
\problem{Proof in $\mathbb{Z}^2$}
The following picture gives the idea for the proof of Blichfeldt's theorem. Explain the picture and complete the proof.
\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}
@ -59,7 +59,7 @@ A region $X$ is \textit{convex} if the line segment connecting any two points in
\definition{}
We say a region is \textit{symmetric with respect to the origin} if for all points $x$ in the region, $-x$ is also in $X$.
We say a region $X$ is \textit{symmetric} if for all points $x \in X$, $-x$ is also in $X$.
\problem{}
\begin{itemize}
@ -74,13 +74,15 @@ Every convex set in $\mathbb{R}^n$ that is symmetric with respect to the origin
\problem{}
Draw a few sets that satisfy \ref{mink} in $\mathbb{R}^2$. Which is the simplest region that has the properties listed above?
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}. Scale this region by $\frac{1}{2}$, called $K' = \frac{1}{2}K$.
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 $K$?