Edits

Advanced/Lattices/parts/1 minkowski.tex

@@ -3,6 +3,11 @@
 \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}
+
+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.}
@@ -33,6 +38,10 @@ The following picture gives the idea for the proof of Blichfeldt's theorem. Expl
 \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{}
@@ -75,7 +84,7 @@ Let $K$ be a region in $\mathbb{R}^2$ satisfying \ref{mink}. Scale this region b
 
 \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 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}