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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, Miriam Schlöter}
\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 4}}
\vspace{15pt}

Please hand in your solutions to Prof.~Ziegler on \textbf{Tuesday, May~20, 2014} before the lecture begins. 

\vspace{20pt}
{\large\bf Problem 1: }\textit{The relative interior of a convex set is convex}\hfill (2+2+2 Points)\\[3pt]
Let us prove the first part of Proposition 3.11 from the lecture, which asserts that for every convex set $K$ the relative interior $\mathrm{relint}(K)$ is also convex. Consider two points $x_0$ and $x_1$ in $\mathrm{relint}(K)$ together with epsilon balls $B_{\epsilon_0}(x_0)$ and $B_{\epsilon_1}(x_1)$ contained in $\mathrm{relint}(K)$. Let $x = \lambda x_0 + (1-\lambda) x_1$ for $\lambda \in (0,1)$ and define $\epsilon := \lambda \epsilon_0 + (1-\lambda) \epsilon_1$. 
\begin{compactenum}[\normalfont (a)]
\item Show that $B_\epsilon(x) \subset \conv\big(B_{\epsilon_0}(x_0) \cup B_{\epsilon_1}(x_1)\big)$. This implies that $x \in \mathrm{relint}(K)$.
\item Show that the choice of $\epsilon$ is maximal in order for the claim in $(a)$ to hold.
\item Show that if $x_0\in\mathrm{relint}(K)$ and $x_2\in K$, the point $\lambda x_0+(1-\lambda)x_2\in\mathrm{relint}(K)$
        for $0<\lambda\le1$.
\end{compactenum}

\vspace{10pt}
{\large\bf Problem 2: }\textit{Intersections of convex sets}\hfill (6 Points)\\[3pt]
Let $n \geq d+1$ and let $A_1,A_2, \dots, A_n$ and $C$ be convex subsets of $\R^d$. Assume that for any $(d+1)$-element index set $I \subseteq \{1, \dots, n\}$ there is a $t_I \in \R^d$ such that
\[
t_I + C \subseteq \bigcap\limits_{i \in I} A_i.
\]
Show that there is a $t_0 \in \R^d$ such that $t_0 + C \subseteq A_i$ for all $i=1, \dots, n.$\\ \emph{Hint:} Do induction on $n$ and apply Radon's Theorem.

\vspace{10pt}
{\large\bf Problem 3: }\textit{Carathéodory's Theorem in a special case}\hfill (8(+2) Points)\\[3pt]
A topological space $X \subseteq \R^d$ is \Emph{path-connected} if any two points $x_0,x_1 \in X$ can be connected by a continuous path, that is, a continuous map $\gamma \colon [0,1] \to X$ such that $\gamma(0)=x_0$ and $\gamma(1) = x_1$.\\
\begin{compactenum}[\normalfont (a)]
\item Let $A$ be a convex body in $\R^d$. Show that if the set of extreme points $\mathrm{ext}(A)$ of $A$ is path-connected, then every point in $A$ lies in the convex hull of $d$ points from $\mathrm{ext}(A)$. This improves the bound of $d+1$ points in Carathéodory's Theorem by~$1$. \emph{Note:} This is not an easy exercise. Do the best you can while still being explicit in your proof.
\item \emph{Bonus:} Is (a) still true if we weaken the assumption that $\mathrm{ext}(A)$ is path-connected and ask that $\mathrm{ext}(A)$ only be connected (the other assumptions remain unchanged)?
\end{compactenum}





\end{document}