\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{soul}
\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{\VV}{\mathcal{V}}

\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 11}}
\vspace{15pt}

Please hand in your solutions to Prof.~Ziegler on \textbf{Wednesday, Jan.~22, 2014} before the lecture begins. 


\vspace{20pt}
{\large\bf Problem 1: }\textit{The Number of Faces: Extremal Properties}\hfill (4(+3) Points)\\
Fix $d$ and let $M_i(n)$ be the maximal and $m_i(n)$ be the minimal number of $i$-faces for a simplicial $d$-polytope with $n$ vertices.
\begin{compactenum}[\normalfont(a)]
\item \begin{compactenum}[(i)]
\item Show that $M_i(n)$ and $m_i(n)$ are polynomials in $n$.
\item Compute the degrees of $M_i(n)$ and $m_i(n)$.
\item Compute the leading coefficients of $M_i(n)$ and $m_i(n)$.
\end{compactenum}
\item \emph{Bonus:} What can be said if the polytope is not required to be simplicial?
\end{compactenum}

\vspace{20pt}
{\large\bf Problem 2: }\textit{M-Sequences in Dimension $4$ and $5$}\hfill (4 Points)\\
Show that $(1,a,b) \in \Z^3$ is an $M$-sequence if and only if
\[
a,b \geq 0 \quad \text{and} \quad b \le \binom{a+1}{2}.
\]

\vspace{20pt}
{\large\bf Problem 3: }\textit{$f$-Vectors of Simplicial Polytopes}\hfill (6 Points)\\
Consider the following potential  $f$-vectors of $4$-polytopes:
\begin{align*}
v_1 &= (1,17,57,80,40) \\
v_2 &= (1,13,84,142,71) \\
v_3 &= (1,17,100,166,83)
\end{align*}
\begin{compactenum}[\normalfont(a)]
\item Compute the corresponding $h$- and $g$-vectors.
\item Which of the $v_i$ is the $f$-vector of a simplicial $4$-polytope?
\end{compactenum}
  


\newpage
{\large\bf Problem 4: }\textit{Counterexample for Motzkin's Unimodality Conjecture}\hfill(6 Points)\\
In the 1950s, T. S. Motzkin and others conjectured that $f$-vectors of convex polytopes are unimodal.  Recall the definition of unimodality from Problem Sheet 6.  A first counterexample was given by A. Björner in 1981~\cite{bjorner1981}. He constructed a simplicial polytope of dimension $24$ with $f_{14} > f_{15} <f_{16}$.

In this exercise you will construct another counterexample: a simplicial polytope $P$ of dimension $30$ whose $f$-vector is not unimodal. You are encouraged to use a computer algebra program such as \emph{Maple} or \emph{Mathematica} to do your calculations.

Let $d=30, n=47$ and $r=65555$. Define
\begin{align*}
g_1 &\coloneqq n - d - 1 + r,\\
g_k &\coloneqq \binom{n-d-2+k}{k}, \quad \text{for } k \ge 2.
\end{align*}
\begin{compactenum}[\normalfont(a)]
\item Check that $g = (1, g_1,\ldots, g_{15})$ is the $g$-vector of a simplicial polytope $P$.
\item Compute the $f$-vector of $P$ and show that it is \emph{not} unimodal.
\item Describe a polytope with this $f$-vector.
\end{compactenum}

\bibliography{DiscGeo11.bib}{}
\bibliographystyle{siam}

\end{document}


