\documentclass[a4paper,twoside,12pt]{article}
%\usepackage [reqno] {amsmath}
\usepackage{amsfonts,amstext}
\usepackage{amsmath}
\usepackage{amstext}
\usepackage[ngerman]{babel}
\usepackage{fullpage}
\usepackage{graphicx}

\newcommand{\ZETTELNUMMER}{4}

\newcounter{AUFGNR}
\setcounter{AUFGNR}{1}
\newcommand{\AUFGABE}[1]{\vspace{0.3cm}\item[Problem \arabic{AUFGNR}]\stepcounter{AUFGNR} #1}


\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}
\newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil}
\newcommand{\half}[1]{\frac{#1}{2}}
\newcommand{\eps}{\varepsilon}

\DeclareMathOperator{\CH}{\text{CH}}
\DeclareMathOperator{\R}{\mathbb{R}}

\renewcommand{\labelenumi}{(\alph{enumi})}
\renewcommand{\labelenumii}{(\roman{enumii})}

\begin{document}
\pagestyle{empty}
\hrule\medskip
\rule{0ex}{0ex}\\[-1ex]
\ZETTELNUMMER th Problem Set for the MDS Block Course on

\medskip
\noindent
\large
\textbf{High-Dimensional Computational Geometry} 

\medskip
\noindent
\normalsize
Helmut Alt, Wolfgang Mulzer\hfill Summer 2013

\medskip\hrule

\bigskip
\noindent
The problems will be discussed in the afternoon session.\\


%\vskip 0.5cm

\begin{description}
\AUFGABE{Area of polygons} 

Derive the formula given in the course for the area of simple polygons (or at least for convex polygons).


\AUFGABE{Volume of convex polytopes}

The volume of a simplex with vertices $v_0, v_1, ... , v_d \in \mathbb{R}^d$ is $1/d!$ times (the absolute value of) the determinant of the $d\times d$-matrix with column vectors $v_i-v_0,$ $i=1,...,d$.

Design and analyze (one or more) algorithms for computing the volume of a general convex (V-)polytope in arbitrary dimension $d$. Your algorithm/s should be as simple as possible, give an exact or approximate result, 
and you may ignore excessive runtimes.

\AUFGABE{Reduction of \#3SAT to \#LEXT}

Prove Lemma 1b).

\end{description}
\end{document}
