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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 6}}
\vspace{15pt}

Please hand in your solutions to Prof.~Ziegler on \textbf{Wednesday, Nov.~27, 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{Unimodality of an $f$-Vector} \hfill (6 Points)\\[5pt]
Let $C_d$ be the $d$-dimensional cube, for instance $C_d = [0,1]^d$.
For which $k=k(d)$ does $C_d$ have the largest number of $k$-faces? 
	%Give an exact answer for small $d$ and an approximate answer for large $d$.
	To answer this, analyze the quotients $f_k/f_{k-1}$, and show that they decrease with $k$.
	Conclude that the $f$-vector of $C_d$ is \Emph{unimodal}, that is, 
	\[
	f_0<f_1<\dots<f_{k(d)}\ge f_{k(d)+1}> \dots> f_{d-1}.
	\] 
\vspace{15pt}

{\large\bf Problem 2: }\textit{The Half-Cube} \hfill (8 Points)\\[5pt]
Let \[
H_d:=\conv \{x\in\{0,1\}^d: x_1+\dots+x_d\in2\Z\}
\] be the $d$-th \Emph{half-cube}.
	\begin{compactenum}[\normalfont (a) ]
		\item Describe the facets of $H_d$: How many are there and what are their combinatorial types?
		(The cases $d=1,2,3$ need to be described separately.)
		\item Give an $\HH$-representation of $H_d$.
		(The cases $d=1,2,3$ need to be argued separately.)
		\item Show that $H_d$ is \Emph{$3$-simplicial}, that is, all of its $3$-faces are tetrahedra.
		\item Show that $H_d$ is simplicial for $d\le4$, but not for $d>4$.
	\end{compactenum}

\vspace{15pt}

\newpage

{\large\bf Problem 3: }\textit{The $f$-Polynomial} \hfill (6 Points)\\[5pt]
Define the \emph{$f$-polynomial} of a $d$-polytope $P$ with $f$-vector $f(P)=(1,f_0,\ldots,f_{d-1},1)$ as
	\[
	f_P(t):= 1 + f_0t+f_1t^2+\dots+f_{d-1}t^d+ t^{d+1}.
	\]
	\begin{compactenum}[\normalfont (a) ]
			\item 
	Describe the $f$-polynomial $f_{P\times I}$ of the prism $P\times I$ in terms of the $f$-polynomial of $P$.
	Deduce from this a formula for the $f$-polynomial of the $d$-cube.
		\item
	Describe the $f$-polynomial of~$P^*$ in terms of the $f$-polynomial of~$P$.
	Deduce from this a formula for the $f$-polynomial of the $d$-crosspolytope.
		\item
	Describe the $f$-polynomial of $P\times Q$ in terms of the $f$-polynomials of $P$ and of~$Q$.
\end{compactenum}

\end{document}
