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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 12}}
\vspace{15pt}

Please hand in your solutions to Prof.~Ziegler on \textbf{Tuesday, July~15, 2014} before the lecture begins. \textcolor{red}{\textbf{This the last sheet. It has 6 bonus points.} }

\vspace{30pt}

{\large\bf Problem 1: }\textit{Mixed Volume of two Ellipses}\hfill (6(+6) Points)\\[3pt]
Let $K$ and $L$ be two ellipses in $\R^2$, say,
\begin{align*}
K &:= \{(x,y) \in \R^2 : \tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} \le 1\}, \\
L &:= \{(x,y) \in \R^2 : \tfrac{x^2}{c^2} + \tfrac{y^2}{d^2} \le 1\}.
\end{align*}
\begin{enumerate}[\normalfont (a) ]
\item Is the Minkowski sum $K+L$ an ellipse?
\item Determine the function
\[
f(\lambda_1, \lambda_2)= \vol_2(\lambda_1 K + \lambda_2 L).
\]
How many values of $f$ do you need in order to determine $f$?
\item\emph{Bonus:} Compute the mixed volume
\[
\mathrm{MV}(K,L).
\]
\end{enumerate}
\vspace{20pt}

{\large\bf Problem 2: }\textit{Mixed Volumes and a One-Dimensional Projection}\hfill (8 Points)\\[3pt]
Let $u \in \R^d \setminus \{0\}$ and let $[0,u]:= \{\lambda u : 0 \le \lambda \le 1\}$. Let $u^\perp$ denote the linear hyperplane perpendicular to $u$. 
\begin{compactenum}[\normalfont (a)]
\item Show that for a convex body $K \subset \R^d$
\[
\MV\big(K[d-1],[0,u]\big) = \frac{\|u \|}{d} \vol_{d-1}\big(\pi_u (K) \big).
\]
Here $\pi_u \colon \R^d \to u^\perp$ is the orthogonal projection onto $u^\perp$.
\item Show that the function $f\colon \R^d \to \R$ given by
\[
f(u):= \MV\big(K[d-1],[0,u] \big)
\]
is convex. [\emph{Hint:} First observe that $f$ is positively linear and then show that $f(u+v) \le f(u) + f(v)$.]
\end{compactenum}

\newpage
% 

{\large\bf Problem 3: }\textit{Volume of a Zonotope}\hfill (8 Points)\\[3pt]
For vectors $z_1,z_2, \dots, z_n \in \R^d \setminus \{0\}$ and using the notation of Problem 2 above define the \Emph{zonotope}
\[
Z := [0,z_1] + [0,z_2] + \dots + [0,z_n].
\]
\begin{compactenum}[\normalfont (a)]
\item Show that \[
\vol_d(Z)\quad = \sum \limits_{1 \le i_1 < i_2 < \dots < i_d \le n } | \det(z_{i_1},z_{i_2}, \dots, z_{i_d}) | \;. 
\]
\item For $\pi_u$ as in Problem 2, consider the function
\[
f(u) := \vol_{d-1}(\pi_u(Z)).
\]
Show that there is a zonotope $\Pi_Z = \sum \limits_{j=1}^M [-w_j, w_j]$ such that 
\[
f(u) = \max \{u^t x: x \in \Pi_Z \} \quad \text{for all } u \in \s^{d-1}.
\]
\end{compactenum}

\end{document}

