\documentclass[a4paper,12pt]{scrartcl}

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

\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 = red,
  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{\C}{\mathbb{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}
\newcommand{\V}{\mathcal{V}}
\DeclareMathOperator{\im}{im}
\newcommand\Stack{\mathrm{Stack}}
\renewcommand{\to}{\longrightarrow}


\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{1.1}

\selectlanguage{german}
\tikzset{
>=stealth',
pil/.style={
           ->,
           shorten <=1pt,
           shorten >=1pt,}
}

\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 5}}
\vspace{15pt}

Please hand in your solutions to Prof.~Ziegler on \textbf{Tuesday, May~27, 2014} before the lecture begins. 

\vspace{20pt}
{\large\bf Problem 1: }\textit{Nearest and extreme points}\hfill (8(+2) Points)
\begin{compactenum}[\normalfont (a)]
\item Let $A\subseteq \R^d$ be a closed set. Recall that a \Emph{nearest point} $x \in A$ to a given point $y \in \R^d$ is a point that minimizes the distance between $y$ and the points of $A$ with respect to the $\ell_2$-norm. In class it was shown that if $A$ is additionally convex, then nearest points are unique. Show the converse: If for every $y \in \R^d \setminus A$ there is a unique nearest point $x=:\pi_A (y) \in A$, then $A$ is convex. 

\item Let $C$ be a convex and compact set and let $H$ be a supporting hyperplane for $C$ at a point $p$ in the boundary $\partial C$. Let $F := H \cap C$ be a ``face''. Show that the set of extreme points $\mathrm{ext} F$ is contained in the set of extreme points $\mathrm{ext} C$. This was a loophole in the proof of Minkowski's Theorem from class.
\item Let $C$ be compact and convex. Show that $C$ has an extreme point. \emph{Note:} Show this from scratch, don't simply refer to the proof of Minkowski's Theorem.
\item \emph{Bonus:} Prove that the set of extreme points of a closed convex set $A \subseteq \R^2$ is closed.
\item Construct an example of a convex and compact set $C$ where the set $\mathrm{ext} C$ is not closed.

\end{compactenum}

\vspace{10pt}
{\large\bf Problem 2: }\textit{Lemma from class}\hfill (6 Points)\\[3pt]
Recall that the \Emph{support function} of a convex body $C \subset \R^d$ was defined as 
\begin{align*}
h_C \colon \R^d &\to \R \\
a &\mapsto \mathrm{max} \{a^t x \colon x \in C\}.
\end{align*}
Prove Lemma 2.31 from class:\\[3pt]
Let $K,L$ and $M$ be convex bodies. Show that
\begin{compactenum}[\normalfont (a)]
\item $h_{K + L}= h_K + h_L$,
\item $ K + M = L + M $ implies that $K = L$.
\end{compactenum}

\vspace{10pt}
{\large\bf Problem 3: }\textit{Polyhedra are spectrahedra}\hfill (6 Points)\\[3pt]
Show that any polyhedron $P$, that is, the interesection of finitely many closed half-spaces in some $\R^d$, is a spectrahedron.
\end{document}