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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 7}}
\vspace{15pt}

Please hand in your solutions to Prof.~Ziegler on \textbf{Wednesday, Dec.~4, 2013} before the lecture begins. \textcolor{blue}{This is the last sheet of the first ``half'' of the semester.}

\vspace{20pt}

{\large\bf Problem 1: }\textit{Joins, Constructing Polytopes} \hfill ((2+)4 Points)\\
\textcolor{blue}{Note: Problem 1 (a) has been corrected and turned into a bonus problem on Mon, Dec.~2. We will discuss it in the tutorials on Wednesday.}
\begin{enumerate}[\normalfont (a) ]
\item \emph{Bonus:} Let $P$ and $Q$ be polytopes each with the origin in its relative interior. For the purpose of this exercise we use the following definition of the join of $P$ and $Q$:
\[
P * Q = \conv \big\{ \left( \ma x \\ 0 \\ 1 \trix \right), \left( \ma 0 \\ y \\ -1 \trix \right) \, : \, x \in P \text{ and } y \in Q \big \}.
\]

 Show that the join construction is self-dual in the following ``strong'' sense: The polar of the join $(P * Q)^*$ is affinely equivalent to the join of the polars $(P^* * Q^*)$.
\item What is the smallest example of a polytope that is not combinatorially equivalent to a non-trivial join, product or direct sum of two polytopes? (Non-trivial means that we exclude cases like the join with the empty set, the product with a point etc.) Before you answer this, state how you interpret ``smallest''.
\end{enumerate}

\vspace{15pt}

{\large\bf Problem 2: }\textit{What can an f-vector tell us?} \hfill (7(+2) Points)\\
Assume for the moment that 
\[
f = f(P) = (1,15,34,28,9,1)
\] 
is the $f$-vector of a polytope $P$.
\begin{compactenum}[\normalfont (a) ]
\item What is the dimension of $P$?
\item Is $P$ simple or simplicial?
\item Could $P$ be a prism?
\item Could $P$ be a pyramid?
\item Could $P$ be a join?
\item Could $P$ be a stacked polytope $\Stack_d(d+1+n)$?
\item \emph{Bonus:} Is $f$ the $f$-vector of a polytope?
\end{compactenum}

\vspace{15pt}

{\large\bf Problem 3: }\textit{Stacked Polytopes and Trees} \hfill (7 Points)\\
Let $d \geq 3$ and $n \geq 3$. To every stacked polytope $\Stack_d(d+1+n)$ we can associate a graph-theoretic tree\footnote{For the definition of a ``tree'' see the excellent Wikipedia article\\ \url{http://en.wikipedia.org/wiki/Tree_(graph_theory)}.} $T = (V,E)$ by adding a vertex $v \in V$ for the simplex $\Delta_d$ and for every simplex we stack onto a facet. We then add an edge $e \in E$ for every two vertices in $V$ which correspond to adjacent simplices. Two simplices are adjacent if one was stacked on a facet of the other.
\begin{enumerate}[\normalfont(a)]
\item Do different trees correspond to different (combinatorial) types of polytopes?
\item Do different types of stacked polytopes $\Stack_d(d+1+n)$ have different trees?
\item Use (i) and (ii) to estimate the number of types of polytopes $\Stack_d(d+1+n)$ for fixed $d \geq 3$ and increasing $n$.
\end{enumerate}

\end{document}
