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\begin{document}
\begin{minipage}[t]{0.5\textwidth}
\begin{flushleft}
\includegraphics[scale=0.09]{bms-logo-eff}\\[5pt]
{\small Prof.~G\"unter M.~Ziegler} \\
{\small Albert Haase, Marie Litz}
\end{flushleft}
\end{minipage}
\begin{minipage}[t]{0.5\textwidth}
\begin{flushright}
\includegraphics[scale=0.4]{fu-logo}\\[5pt]
{\small Institut f"ur Mathematik }\\
{\small Arbeitsgruppe Diskrete Geometrie}\\
\end{flushright}
\end{minipage}

\vspace{20pt}
\centerline{\Large \bf{Discrete Geometry 1 -- Problem Sheet 12}}
\vspace{15pt}

Please hand in your solutions to Prof.~Ziegler on \textbf{Wednesday, Jan.~29, 2014} before the lecture begins. 


\vspace{20pt}
{\large\bf Problem 1: }\textit{Points in Convex Position}\hfill (2+6 Points)\\
Recall that a set $X \subset \R^d$ is \Emph{in convex position} if for every $x \in X$ we have $x \notin \conv(X {\setminus} \{x\})$. 
\begin{compactenum}[\normalfont(a)]
\item Find a configuration of $8$ points in general position in the plane such that \emph{no} 5 of its points are in convex position. Hence you are showing that the ``Erd{\H{o}}s--Szekeres number'' $n(5) > 8$.
%\item \emph{Bonus:} Determine $n(5)$.
\item Prove that for each $k \ge 1$ there exists a number $m(k)$ such that any  $m(k)$ points in the plane contain $k$ points in convex position or $k$ points on a line.
\end{compactenum}
\vspace{20pt}

{\large\bf Problem 2: }\textit{Dual Configurations}\hfill (4+4 Points) 
\begin{compactenum}[\normalfont (a)]
\item Translate the Sylvester--Gallai Theorem into the dual setting of line arrangements. (In your statement, be careful about parallel lines etc.)
\item 
Translate the Erd{\H{o}}s--Szekeres Theorem into the dual setting of line arrangements. 
What is the dual statement to existence of $k$-caps or $k$-cups in every sufficiently large point configuration in general position?
\end{compactenum}
\vspace{20pt}

{\large\bf Problem 3: }\textit{Hyperplane Arrangements} \hfill (4(+6) Points)
\begin{compactenum}[\normalfont (a)]
\item Consider the   arrangement $\mathcal{H}$ of hyperplanes given by the equations $x_i = x_j$ for $1 \le i < j \le d$. Draw a picture of $\mathcal{H}$ in dimension $3$. How many $d$-dimensional cells does $\mathcal{H}$ have?
\item \emph{Bonus:} Consider the   arrangement $\mathcal{H}'$ of hyperplanes given by the equations $x_i = \pm x_j$ for $1 \le i < j \le d$. Draw a picture of $\mathcal{H}'$ in dimensions $2$ and $3$. How many $d$-dimensional cells does $\mathcal{H}'$ have?
%Consider the related arrangement $\mathcal{G}$ of hyperplanes given by the equations $x_i - x_j \in\{0,1,-1\}$ for $1 \le i < j \le d$. Draw a picture of $\mathcal{G}$ in dimension~$2$. How many $d$-dimensional cells does $\mathcal{G}$ have?
\end{compactenum}

\end{document}


