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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 6}}
\vspace{15pt}

Please hand in your solutions to Prof.~Ziegler on \textbf{Tuesday, June~03, 2014} before the lecture begins. 

\vspace{20pt}
{\large\bf Problem 1: }\textit{Convex bodies, polars, and ellipsoids}\hfill (8(+2) Points)\\[3pt]
Given a set $A \subseteq \R^d$, the \Emph{polar} of $A$, denoted by $A^*$, is defined as
\[
A^* := \{ y \in \R^d \colon \langle y, x \rangle \le 1 \text{ for all } x \in A \}.
\]
Let $K$ be a (full-dimensional) convex body containing the origin in its interior.
\begin{compactenum}[\normalfont (a)]
\item Show that $K^*$ is a convex body containing the origin in its interior.
\item Show that $(K^*)^* = K$.
\item Let $E$ be an ellipsoid centered at the origin given by $E = \{x \in \R^d \colon \langle Qx, x \rangle \le 1 \}$, where $Q$ is a positive definite matrix. Show that $E^* =  \{y \in \R^d \colon \langle Q^{-1}y, y \rangle \le 1 \}.$
\item \emph{Bonus:} Show that the image of an ellipsoid under a surjective affine map is again an ellipsoid.
\end{compactenum}

\vspace{20pt}

{\large\bf Problem 2: }\textit{Löwner--John: maximal volume ellipsoid}\hfill (12(+2) Points)
\begin{compactenum}[\normalfont (a)]
\item Let $\Delta_d$ be the standard $d$-simplex given by
\[
\Delta_d := \{ x= (x_1, \dots, x_{d+1}) \in \R^{d+1} \colon x_i \geq 0 \text{ and } x_1 + \dots x_{d+1} = 1\}.
\]
View $\Delta_d$ as a $d$-dimensional convex body lying in the affine subspace given by the hyperplane $H := \{x \in \R^d \colon x_1 + \dots + x_{d+1} = 1 \}.$ Determine the ellipsoid $E_1$ of maximal volume contained in $\Delta_d$. If $c$ is the center of $E_1$, show that the constant $\rho = d$ is minimal for the inclusion $\Delta_d - c \subseteq \rho  (E_1 - c)$ to hold.\footnote{Given a set $X \subset \R^d$, the expression $X - c$ denotes the set $\{ x - c \colon x \in X\}$. Given a positive constant $\delta$, the expression $\delta X$ refers to the set $\{\delta x \colon x \in X\}$.} 
\item Let $C_d$ be the $d$-cube given by
\[
C_d := \{ (x_1, \dots, x_d) \in \R^d \colon |x_i | \le 1 \}.
\]
Determine the ellipsoid $E_2$ of maximal volume contained in $C_d$.
\item Let $C^*_d$ be the standard $d$-dimensional octahedron given by
\[
C^*_d := \{(x_1, \dots, x_d) \in \R^d \colon |x_1 | + \dots + |x_d| \le 1 \}.
\]
Determine the ellipsoid $E_3$ of maximal volume contained in $C^*_d$. Prove that $\rho = \sqrt{d}$ is minimal for the inclusion $C^*_d \subseteq \rho E_3$ to hold.
\item \emph{Bonus:} Let $C$ be a convex polytope such that its symmetry group $G$ acts transitively on its set of vertices. Show that the center of the ellipsoid $E$ of maximal volume contained in $C$ coincides with the barycenter of the vertex set of $C$.\footnote{The action of a group $G$ on a set $X$ is \Emph{transitive} if there is only one orbit, that is, if $Gx = \{gx \colon g \in G\} = X$ for $x \in X$. The barycenter of a set of points $\{x_1, \dots, x_n\} \subset \R^d$ is the point $\sum_i \frac{1}{n}x_i$.} Does this imply that $E$ is a ball?
\end{compactenum}

\end{document}