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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 8}}
\vspace{15pt}

Please hand in your solutions to Prof.~Ziegler on \textbf{Tuesday, June~18, 2014} before the lecture begins. 

\vspace{30pt}

{\large\bf Problem 1: }\textit{Equality}\hfill (6 Points)\\[3pt]
Let $K$ and $L$ be $d$-dimensional polyboxes in $\R^d$, that is, nonempty finite unions of $d$-dimensional axis-parallel rectangular boxes. Let's say $K$ and $L$ are positively homothetic, that is, there is a $\lambda >0$ such that $L = \lambda K + c$ for some $c \in \R^d$. Is the Brunn-Minkowski Inequality 
\[
\sqrt[d]{\vol_d(K+L)} \geq \sqrt[d]{\vol_d(K)} + \sqrt[d]{\vol_d(L)}
\]
 satisfied with equality?  
 
\vspace{30pt}

{\large\bf Problem 2: }\textit{Multiplicative Brunn--Minkowski}\hfill (8 Points)
\begin{compactenum}[\normalfont (a)]
\item Prove the \emph{weighted} arithmetic-geometric-mean  inequality
\[
\sum\limits_{i=1}^n \lambda_i z_i \geq \prod\limits_{i=1}^n z_i^{\lambda_i}
\]for all $z_i  \geq 0$ and $\lambda_i \in [0,1]$ such that $\sum_{i=1}^n \lambda_i =1$.
\item Prove the \emph{multiplicative} Brunn--Minkowski inequality
\[
\vol_d((1-\lambda)K + \lambda L ) \geq \vol_d(K)^{1-\lambda} \, \vol_d(L)^\lambda
\]
for all convex bodies $K,L \subset \R^d$ and $\lambda \in [0,1]$.
\item What are the equality cases?
\end{compactenum}

\newpage

{\large\bf Problem 3: }\textit{Cayley Embedding}\hfill (6 Points)\\[3pt]
Let $K_1, \dots, K_n \subset \R^d$ be convex bodies. The \Emph{Cayley embedding} of $(K_i: i=1, \dots, n)$ is defined as 
\[
C := \Cay(K_1,K_2,\dots,K_n) := \conv\{ (p_i,e_i) \; : \; p_i \in K_i, \; i=1, \dots, n \} \subset \R^d \times \R^n,
\]
where $e_i$ denotes the $i$-th unit vector in $\R^n$. (The Cayley embedding for two sets was used in the proof of Brunn's Slice Inequality.)
\begin{compactenum}[\normalfont (a)]
\item Let $\lambda=(\lambda_1,\dots,\lambda_n) \in \R^n$ with $\lambda_i \geq 0$ for $i=1, \dots, n$ and $\sum_i \lambda_i =1$. Show that
\[
C \cap \{(x,y) \; : \; y = \lambda \} \cong \sum\limits_{i=1}^n \lambda_i K_i.
\]
\item Show that for every proper subset $I \subset [n]$ the Cayley embedding $\Cay(K_i : i \in I)$ is a face of $C$.
\item For $m \geq n$, let $P \subset \R^m$ be an $m$-polytope and let $\pi: \R^m \to \R^n$ be an affine map. Show that if $Q:= \pi(P)$ is an $(n-1)$-simplex and every vertex of $P$ is mapped to a vertex of $Q$, then $P$ is affinely isomorphic to $\Cay(P_1,\dots,P_n)$ for some polytopes $P_1,\dots, P_n$.
\end{compactenum}



\end{document}