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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{20pt}
{\large\bf Problem 1: }\textit{Volume of the free sum}\hfill (8 Points)\\[3pt]
We will verify Proposition 2.61 part (3) from class in a special case. Let $K$ and $L$ both be simplicial convex polytopes of dimensions $k$ respectively $\ell$ with the origin in their interiors. Prove that
\[
\mathrm{vol}_{k+\ell}(K \oplus L) \; = \; \binom{k+\ell}{k}^{-1}\mathrm{vol}_{k}(K) \; \mathrm{vol}_{\ell}(L).
\]

\vspace{20pt}

{\large\bf Problem 2: }\textit{Volume of cross sections}\hfill (6 Points)\\[3pt]
Let $C_3=[0,1]^3$ be the $3$-dimensional unit cube. For $t \in \R$, define the hyperplane
\[
H_t := \{ x \in \R^3 \colon x_1+x_2+x_3 = t\}.
\]
Now define a function $f \colon \R \to \R$ by
\[
f(t) := \mathrm{vol}_2(C_3 \cap H_t).
\]
Explicitly express the function $f$  in terms of $t$ and sketch the graphs of $t \mapsto f(t)$ and of $t \mapsto \sqrt{f(t)}$.
\vspace{20pt}

{\large\bf Problem 3: }\textit{Signed volume}\hfill (6 Points)\\[3pt]
For a triangle $\Delta = \Delta(a,b,c) = \conv\{a,b,c\}$ with ordered vertices $a,b,c \in \R^2$ we define the \Emph{signed volume} of $\Delta$ as
\[
\mathrm{vol}_2^\circ(\Delta) \; := \; \frac{1}{2} \, \det \begin{pmatrix} 1 &1 &1 \\ a & b & c \end{pmatrix}.
\]
Let $P = \conv \{v_1, \dots, v_n \}$ be a polygon with vertices $v_i \in \R^2$ ordered counterclockwise. Let $p \in \mathrm{int}(P)$ be a point in the interior of $P$. 
\begin{compactenum}[\normalfont (a)]
\item Show the following equality where $\mathrm{vol}_2(P)$ denotes the standard $2$-dimensional volume of $P$:
\[
\mathrm{vol}_2(P) = \frac{1}{2} \sum\limits_{i=1}^n \mathrm{vol}_2^\circ(\Delta(p,v_i, v_{i+1})) \quad \text{where } v_{n+1} := v_1. \tag{$*$}
\]
\item Argue geometrically that the right-hand side of ($*$) is independent of the choice of $p \in \R^2$ --- whether $p \in P$ or not. \\
(This shows that the volume is a polynomial of  the vertex coordinates and invariant under translation).
\end{compactenum}
\end{document}