\documentclass[a4paper,12pt]{scrartcl}

% \usepackage[english,german]{babel} % [verfuegbare Sprachen, letzgenannte ist aktiv]

\usepackage{graphicx} % include eps graphics
\usepackage{psfrag} % .eps properly aligned, scaled, and rotated
\usepackage{wrapfig} % WRAP FIGURE [text umfliest das Bild]
\usepackage[ngerman,english]{babel}
\usepackage[utf8]{inputenc}
\usepackage{hyperref}
\usepackage{mathtools}
\usepackage{ngerman} % Silbentrennung ect. in neuer-Rechtschreibung-deutsch
\usepackage{eurosym} % Das euro Symbol [nicht mehr]
\usepackage{color} % textcolors [RGB oder auch 68 Standardfarben]
\usepackage{enumerate} % aufzaehlungen
\usepackage{paralist} % kuerzere Aufzaehlungen
\usepackage{url} % bessere Darstellung (hyperref: auch anklickbar)
\usepackage{alltt} % ermoeglicht uninterpretierten Text (sozusagen String)
\usepackage{ae,aecompl} % ecFonts auf cmFonts zurueckfuehren
\usepackage{ifthen}
\usepackage{amsmath} % MATH allignment[Matrix,Brueche,Gleichungen]
\usepackage{amssymb} % Mathe SYMBOLE
\usepackage{amsthm} % Mathe THEOREM [theorem,lemma,proof] Gliederung
\usepackage{verbatim} % multiline comment
\usepackage{paralist}
\parindent =0pt
\usepackage{geometry}
\graphicspath{{../../figures/}}
\geometry{
  body={},
  left=1.2in,
  right = 1.2 in,
  top=1.25in
}
\hypersetup{
  colorlinks = true,
  urlcolor = blue,
  pdfauthor = {},
  pdfkeywords = {albert haase, mathematics, mathematik},
  pdfsubject = {},
  pdfpagemode = UseNone
}

\newcommand{\R}{\mathbb{R}}
\newcommand{\D}{\mathbb{D}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\s}{\mathbb{S}}
\newcommand{\su}{\mathcal{S}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\ti}{\mathcal{T}}
\newcommand{\cc}{\mathcal{C}}
\newcommand{\dd}{\mathcal{D}}
\newcommand{\E}{\mathbf{E}}
\newcommand{\HRule}{\rule{1.03\textwidth}{0.5mm}}
\renewcommand\O{\mbox{\rm O}}
%\renewcommand{\subset}{\subseteq}
\renewcommand{\H}{\mathcal{H}}
\newcommand{\V}{\mathcal{V}}
\DeclareMathOperator{\im}{im}
\newcommand\Stack{\mathrm{Stack}}


\newcommand{\red}{\textcolor{red}} %Farben
\newcommand{\green}{\textcolor{green}}
\newcommand{\ink}{\textcolor[rgb]{0,.2,.8}}
\newcommand{\purp}{\textcolor[rgb]{0.9,0,.5}}
\newcommand{\sky}{\textcolor[rgb]{0,.5,.7}}
\newcommand{\tree}{\textcolor[rgb]{0,.4,.1}}
\newcommand\Emph[1]{\emph{\color{blue}#1}}
\renewcommand{\epsilon}{\varepsilon}
\newcommand{\HH}{\mathcal{H}}


\newcommand{\set}[1]{\ensuremath{ \left\{ #1 \right\}}}
\newcommand{\mset}[2]{\ensuremath{\left\{ #1  :  #2 \right\}}}
\renewcommand{\mapsto}{\longmapsto}
\newcommand{\inner}[1]{\left\langle \, #1  \,\right\rangle}
\newcommand{\linspan}[1]{\ensuremath{\text{span}\left\{ #1 \right \} }}
\newcommand{\ma}{\begin{smallmatrix} } \newcommand{\trix}{\end{smallmatrix}} % Matrix
\newcommand\aff{\operatorname{aff}}
\newcommand\conv{\operatorname{conv}}
\DeclareMathOperator{\cone}{cone}
\DeclareMathOperator{\inter}{int}
\linespread{1.3}

\selectlanguage{german}

\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 8}}
\vspace{15pt}

Please hand in your solutions to Prof.~Ziegler on \textbf{Wednesday, Dec.~11, 2013} before the lecture begins. 

\vspace{20pt}

{\large\bf Problem 1: }\textit{$k$-Neighborliness} \hfill (6 Points)\\
Let $P$ be a $k$-neighborly polytope of dimension $d \geq 1$. Show that every $(2k-1)$-face of $P$ is a simplex. Conclude that if $P$ is $(\lfloor \frac{d}{2} \rfloor +1)$-neighborly, then $P$ is a simplex.

\vspace{15pt}

{\large\bf Problem 2: }\textit{Cyclic Polytopes} \hfill (4(+2)+4 Points)\\
Consider the cyclic polytope
\[
C_d(t_1, \ldots, t_n) \coloneqq \conv \set{\gamma(t_1), \ldots, \gamma(t_n)} \subset \R^d,
\]
where $t_1 < \cdots < t_n$ for $n > d > 1$ and $\gamma$ denotes the moment curve in $\R^d$. In this problem we are referring to combinatorial types rather than geometric realizations.
\begin{enumerate}[\normalfont (a) ]
\item Let $d$ be even.  Let $\pi = (1\,2\,\ldots n)$ be the cyclic permutation that sends $i \mapsto i+1$ for $1 \le i \le n-1$ and $n \mapsto 1$. Show that whenever $I \subset \set{1,\ldots,n}$ is the index set (of the vertices) of a face of $C_d(t_1,\ldots, t_n)$ the set $\pi(I)$ is an index set of a face of $C_d(t_1,\ldots, t_n)$. Prove the same result for the permutation $\pi'$ that sends $i \mapsto n+1 -i $, in other words, for
\[
\pi'=\prod\limits_{\substack{i < j \\ i + j = n+1}} (i,j).
\]
\item \emph{Bonus:} Do the statements in (a) hold for odd $d$? 
\item Show that $C_d(d+1)$ is a $d$-simplex $\Delta_d$ and that $C_d(d+2)$ is the direct sum of simplices $\Delta_{\lceil d/2 \rceil} \oplus \Delta_{\lfloor d/2 \rfloor}$.
\end{enumerate}

\newpage

{\large\bf Problem 3: }\textit{Examples} \hfill (3+3 Points)
\begin{enumerate}[\normalfont (a) ]
\item Give an example of a (geometric) polytope $P$ of dimension 4 with 1000 facets such that every two facets share a 2-face. By ``share a 2-face'' we mean that $P$ has a $2$-face incident to both facets.
\item Give an example of a simplicial polytope of dimension $5$ with $7$ vertices whose combinatorial type is not that of a cyclic polytope $C_5(7)$.
\end{enumerate}


\end{document}
