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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 10}}
\vspace{15pt}

Please hand in your solutions to Prof.~Ziegler on \textbf{Wednesday, Jan.~15, 2014} before the lecture begins. 

\vspace{20pt}
{\large\bf Problem 1: }\textit{Calculate an $h$-Vector}\hfill (6 Points)\\
Let $C_d^{\Delta}$ denote the $d$-dimensional crosspolytope. Find and describe a suitable shelling of $\mathcal{C} (\partial C_d^{\Delta})$ and compute the $h$-vector of $C_d^{\Delta}$.  

\vspace{15pt}

{\large\bf Problem 2: }\textit{Dehn-Sommerville Equations}\hfill (2+2 Points)\\
Let $P$ be a simplicial $d$-polytope.
\begin{compactenum}[\normalfont (a) ]
\item Check that the Dehn-Sommerville equations for $d=4$ are
equivalent to the two linear relations $f_0(P) - f_1(P) + f_2(P) - f_3(P) = 0$ and $f_2(P) = 2f_3(P)$.
\item For $d=5$, find a linear relation that follows from the Dehn-Sommerville
equations but is independent of the Euler-Poincar\'e relation and $2f_3(P) = 5f_4(P)$.
\end{compactenum}

\vspace{15pt}

{\large\bf Problem 3: }\textit{$f$-Vectors of Neighborly Polytopes}\hfill (2+2+2 Points)
\begin{compactenum}[\normalfont (a) ]
\item Compute the $f$-vector of the cyclic polytope $C_4(7)$.
\item Prove the following statement: The $f$-vectors of all neighborly simplicial polytopes of dimension $d$ on $n$ vertices are identical.
\item What is the maximal possible number of vertices in a simple $5$-polytope with $8$ facets?
\end{compactenum}

\vspace{15pt}

{\large\bf Problem 4: }\textit{Shellings for the Cube}\hfill (3+3 Points)
\begin{compactenum}[\normalfont (a) ]
\item How many different shellings are there for the (boundary of the) $3$-dimensional cube $C_3$?
\item Show (using induction on the dimension $d$) that a facet ordering $F_1,\dots,F_{2d}$ of the $d$-cube $C_d$  is \emph{not} a shelling if and only if $F_1,\dots,F_{2j}$ consists of $j$ pairs of opposite facets, for some $j$, $1\le j<d$.
\end{compactenum}


\end{document}
