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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 -- Xmas Problem Sheet}}
\vspace{15pt}

These are bonus problems for the Xmas break -- have fun with them.

If you want them graded or would like credit, please hand in your solutions to Dr.~Izmestiev on \textbf{Wednesday, Jan.~8, 2014} before the lecture begins. (We'll count the bonus points for the first or second half of the semester, as you like. Please specify.)

\vspace{20pt}

{\large\bf Problem 1: }\textit{Triangles or simple vertices }\hfill ((4) Points)\\
Show that every $3$-dimensional polytope has at least $4$ triangle faces or at least $4$ simple vertices (i.e., vertices of degree $3$) or both.
Show that $3$-polytopes with only $4$ triangle faces and exactly $4$ simple vertices but an arbitrarily large number of edges 
exist.

\bigskip
{\large\bf Problem 2: }\textit{$f$-vectors of $3$-polytopes}\hfill \hfill ((6)+(6) Points)
%; Steinitz 1906 \cite{Stei3}]
        \begin{compactenum}[(i)]
                \item
        Show that the $f$-vectors of $3$-polytopes are given by the set
        \[
        %\mathcal{F}_3=
        \{ (f_0,f_1,f_2)\in\Z^3 : f_0-f_1+f_2=2,\quad
                                  f_2 \le 2 f_0-4,\quad
                                  f_0 \le 2 f_2-4\}.
        \]
        % \item
        %   Show that the $f$-vectors of the inscribable $3$-polytopes are also given by $\mathcal F$.
        \item
        Show that the $f$-vectors of centrally-symmetric $3$-polytopes are given by  
        \[
        \mbox{\hspace{-10mm}}\{ (f_0,f_1,f_2)\in(2\Z)^3 : f_0-f_1+f_2=2,\quad
                                  f_2 \le 2 f_0-4,\quad
                                  f_0 \le 2 f_2-4,\quad
                                  f_0+f_2\ge14\}.
        \]
        \end{compactenum}


\bigskip
{\large\bf Problem 3: }\textit{The ``fractional cube slice polytopes'' }\\% \hfill (17(+4) Points)\\
For an odd integer $\ell$, $1\le \ell\le 2d-1$ let 
    \begin{align*}
        \Delta_{d-1}(\tfrac\ell2):= &\ \{x\in[0,1]^d:x_1+\dots+x_d=\tfrac\ell2\}.
    \end{align*}
    \begin{compactenum}[(i)]
        \item Show that $\Delta_{d-1}(k)$ is affinely equivalent to \hfill ((2) Points)
            \begin{align*}
            \Delta'_{d-1}(\tfrac\ell2):= &\ \{x\in[0,1]^{d-1}:\tfrac\ell2-1\le x_1+\dots+x_{d-1}\le \tfrac\ell2\}.
            \end{align*}
            \item Describe $\Delta_2(\frac32)$. \hfill ((1) Point)
            \item Describe $\Delta_3(\frac32)$. (It is a \emph{semi-regular} polytope: Its faces are regular polygons,
                and at each vertex it has the same number and types of faces.) \hfill ((1) Point)
        \item Study how the hyperplane $H_{\ell/2}=\{x\in\R^d:x_1+\dots+x_d=\frac\ell2\}$ cuts the faces of 
            the $d$-cube $[0,1]^d$. What do the resulting faces look like? Conversely, describe how the faces of $\Delta_{d-1}(\frac\ell2)$ arise from faces of~$[0,1]^d$. \hfill  ((2) Points)
        \item Give a $\VV$-description of $\Delta_{d-1}(\frac\ell2)$.\hfill  ((1) Point)
        \item Show that $\Delta_{d-1}(\frac\ell2)$ and $\Delta_{d-1}(d-\frac\ell2)$ are congruent. 
          Derive that for odd $d$, $\Delta_{d-1}(\frac d2)$ is {centrally symmetric}. \hfill  ((1) Point)
        \item For $\Delta_{d-1}(\frac12)$ and $\Delta_{d-1}(\frac{2d-1}2)$ are simplices. 
         Show that for $3\le\ell\le2d-3$, $\Delta_{d-1}(\frac\ell2)$ has $2d$ facets. What are their combinatorial types? There are two different combinatorial types, except in the case $\ell=d$, for odd $d$. \hfill ((2) Points)
        \item  State and prove a formula for the $f$-vector of $\Delta_{d-1}(\frac\ell2)$.  \hfill ((4) Points)
         \item Compute and plot the $f$-vectors of
          $\Delta_{42}(\frac{13}2)$, $\Delta_{42}(\frac{23}2)$, $\Delta_{42}(\frac{33}2)$, and $\Delta_{42}(\frac{43}2)$. \\
          \mbox{}\hfill ((2) Points)
    \end{compactenum}
       
\end{document}
