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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}

\enlargethispage{5mm}
\vspace{20pt}
\centerline{\Large \bf{Discrete Geometry II -- Problem Sheet 3}}
\vspace{15pt}

Please hand in your solutions to Prof.~Ziegler on \textbf{Tuesday, May~13, 2014} before the lecture begins. 

\vspace{20pt}
{\large\bf Problem 1: }\textit{Convex hulls}\hfill (6 Points)
\begin{compactenum}[\normalfont (a)]
\item Give an example of a closed subset of $\R^2$ whose convex hull is not closed. 
\item Prove that the convex hull of an open set in $\R^d$ is open.
\end{compactenum}

\vspace{20pt}

{\large\bf Problem 2: }\textit{Convex bodies}\hfill (8 Points)\\[3pt]
Recall that a subset $C \subset \R^d$ is called a \Emph{convex body} if it is convex, compact, and full-dimensional.
\begin{compactenum}[\normalfont (a)]
\item Determine the cardinality of the set of convex bodies in $\R^2$.
\item Show that there is a sequence $K_1, K_2, \dots$ of convex bodies in $\R^2$ such that for every $\epsilon > 0$ and for each convex body $C \subset \R^2$, there is some $i$ such that $K_i \subseteq C$ and $\mathrm{vol}(K_i) \geq(1-\epsilon) \mathrm{vol}(C)$.\footnote{To be more precise, we could say that ``$\mathrm{vol}$'' is the two-dimensional Lebesgue measure.}
\end{compactenum}

% \vspace{20pt}
% 
% {\large\bf Problem 3: }\textit{Roots of the derivative}\hfill (6 Points)\\[3pt]
% Identify the complex numbers $\C$ with $\R^2$ and let $p \in \C[z]$ be a complex polynomial in one variable with roots $r_1, \dots, r_d$. Show that the roots of the derivative $p'(z)$ are contained in $\mathrm{conv}\{r_1, \dots, r_d\}$. 

\vspace{20pt}
{\large\bf Problem 3: }\textit{Extreme and exposed points}\hfill (6 Points)\\[3pt]
Let $C \subseteq \R^d$ be a non-empty convex set. 
\begin{compactenum}[\normalfont (a)]
\item A point $z \in C$ is called an \Emph{extreme point} of $C$ if for every 
$x,y \in C$, $z~\in~\mathrm{conv}\{x,y\}$ if and only if $z = x$ or $z=y$.

Show that $z$ is an extreme point of $C$ if and only $C \setminus \{z\}$ is convex. Infer that if $X$ is a finite set and $z \in \mathrm{conv}(X)$ is an extreme point, then $z \in X$.
\item A point $z \in C$ is called an \Emph{exposed point} of $C$ if there is a vector $c\in\R^d$ such that $c^tz>c^tx$ for all $x \in C {\setminus} \{z\}$.

Show that every exposed point is extreme, but extreme points are not exposed in general.

\item %Let $C$ additionally be compact and let 
     Let $z \in C$ be a point such that $\| z \|_p \geq \| y \|_p$ for some $1<p<\infty$ and for all $y \in C$. Show that $z$ is an exposed point of $C$.
\end{compactenum}
\end{document}
