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\begin{document}
\begin{minipage}[t]{0.5\textwidth}
\begin{flushleft}
\includegraphics[scale=0.09]{bms-logo-eff.pdf}\\[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 3}}
\vspace{15pt}

Please hand in your solutions to Prof.~Ziegler on \textbf{Wednesday, Nov.~6, 2013} before the lecture begins. Please put your name and student ID (if you have one) on the first page of your solutions and staple the sheets together.

\vspace{20pt}

{\large\bf Problem 1: }\textit{Posets} \hfill (4 Points)\\[5pt]
Let $(Q, \le )$ be a finite partially ordered set (poset). 
Show that any two of the following three properties yield the third property:
\begin{enumerate}[\normalfont (i) ]
\item $(Q, \le )$ is bounded.
\item For any two elements $a,b \in Q$ their join $a \vee b$ exists.
\item For any two elements $a,b \in Q$ their meet $a \wedge b$ exists.
\end{enumerate}


\vspace{20pt}

{\large\bf Problem 2: }\textit{Faces and Lattices} \hfill (4+4 Points)\\[5pt]
Recall that the set of faces $F$ of a polytope $P$ together with the inclusion relation ``$\subset$'' is a finite graded lattice that satisfies the diamond property.
\begin{enumerate}[\normalfont (a) ]
\item Give an example of a finite graded lattice satisfying the diamond property that is not the face lattice of a polytope. Give an explanation.
\item Let $P$ be a polytope with vertex set $V$. Prove that any subset of $V$ defines a unique face of $P$ if and only if $P$ is a simplex.
\end{enumerate}
\vspace{20pt}

{\large\bf Problem 3: }\textit{Example of a Face Lattice} \hfill (2+2+4(+2) Points)\\[5pt]
Consider the example of a face lattice $L(P)$ of a polytope $P$ with vertices $V= \set{v_0, \ldots, v_9}$ that Professor Ziegler handed out in class on Wednesday. It can be found on the course website under ``Lecture Notes Etc.''.\\

Answer the following questions and explain how you came up with your answers.
\begin{enumerate}[\normalfont (a) ]
\item How many vertices do the $2$-faces of $P$ have?
\item How many facets do the $3$-faces of $P$ have?
\item Choose three vertices $u,v,w \in V$ of $P$ and determine the number of vertices that the vertex figures $P/u, P/v$ and $P/w$ have.
\item \emph{Bonus:} Choose a vertex Figure $P/v$ for $v \in V$ and draw a polytope that has the same combinatorial type as $P/v$. (The \Emph{combinatorial type} of a polytope is the equivalence class of its face lattice under isomorphism.)
\end{enumerate}
\end{document}
